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rl/Lib/site-packages/sympy/polys/agca/__init__.py

@@ -0,0 +1,5 @@
+"""Module for algebraic geometry and commutative algebra."""
+
+from .homomorphisms import homomorphism
+
+__all__ = ['homomorphism']

rl/Lib/site-packages/sympy/polys/agca/extensions.py

@@ -0,0 +1,356 @@
+"""Finite extensions of ring domains."""
+
+from sympy.polys.domains.domain import Domain
+from sympy.polys.domains.domainelement import DomainElement
+from sympy.polys.polyerrors import (CoercionFailed, NotInvertible,
+        GeneratorsError)
+from sympy.polys.polytools import Poly
+from sympy.printing.defaults import DefaultPrinting
+
+
+class ExtensionElement(DomainElement, DefaultPrinting):
+    """
+    Element of a finite extension.
+
+    A class of univariate polynomials modulo the ``modulus``
+    of the extension ``ext``. It is represented by the
+    unique polynomial ``rep`` of lowest degree. Both
+    ``rep`` and the representation ``mod`` of ``modulus``
+    are of class DMP.
+
+    """
+    __slots__ = ('rep', 'ext')
+
+    def __init__(self, rep, ext):
+        self.rep = rep
+        self.ext = ext
+
+    def parent(f):
+        return f.ext
+
+    def as_expr(f):
+        return f.ext.to_sympy(f)
+
+    def __bool__(f):
+        return bool(f.rep)
+
+    def __pos__(f):
+        return f
+
+    def __neg__(f):
+        return ExtElem(-f.rep, f.ext)
+
+    def _get_rep(f, g):
+        if isinstance(g, ExtElem):
+            if g.ext == f.ext:
+                return g.rep
+            else:
+                return None
+        else:
+            try:
+                g = f.ext.convert(g)
+                return g.rep
+            except CoercionFailed:
+                return None
+
+    def __add__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem(f.rep + rep, f.ext)
+        else:
+            return NotImplemented
+
+    __radd__ = __add__
+
+    def __sub__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem(f.rep - rep, f.ext)
+        else:
+            return NotImplemented
+
+    def __rsub__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem(rep - f.rep, f.ext)
+        else:
+            return NotImplemented
+
+    def __mul__(f, g):
+        rep = f._get_rep(g)
+        if rep is not None:
+            return ExtElem((f.rep * rep) % f.ext.mod, f.ext)
+        else:
+            return NotImplemented
+
+    __rmul__ = __mul__
+
+    def _divcheck(f):
+        """Raise if division is not implemented for this divisor"""
+        if not f:
+            raise NotInvertible('Zero divisor')
+        elif f.ext.is_Field:
+            return True
+        elif f.rep.is_ground and f.ext.domain.is_unit(f.rep.LC()):
+            return True
+        else:
+            # Some cases like (2*x + 2)/2 over ZZ will fail here. It is
+            # unclear how to implement division in general if the ground
+            # domain is not a field so for now it was decided to restrict the
+            # implementation to division by invertible constants.
+            msg = (f"Can not invert {f} in {f.ext}. "
+                    "Only division by invertible constants is implemented.")
+            raise NotImplementedError(msg)
+
+    def inverse(f):
+        """Multiplicative inverse.
+
+        Raises
+        ======
+
+        NotInvertible
+            If the element is a zero divisor.
+
+        """
+        f._divcheck()
+
+        if f.ext.is_Field:
+            invrep = f.rep.invert(f.ext.mod)
+        else:
+            R = f.ext.ring
+            invrep = R.exquo(R.one, f.rep)
+
+        return ExtElem(invrep, f.ext)
+
+    def __truediv__(f, g):
+        rep = f._get_rep(g)
+        if rep is None:
+            return NotImplemented
+        g = ExtElem(rep, f.ext)
+
+        try:
+            ginv = g.inverse()
+        except NotInvertible:
+            raise ZeroDivisionError(f"{f} / {g}")
+
+        return f * ginv
+
+    __floordiv__ = __truediv__
+
+    def __rtruediv__(f, g):
+        try:
+            g = f.ext.convert(g)
+        except CoercionFailed:
+            return NotImplemented
+        return g / f
+
+    __rfloordiv__ = __rtruediv__
+
+    def __mod__(f, g):
+        rep = f._get_rep(g)
+        if rep is None:
+            return NotImplemented
+        g = ExtElem(rep, f.ext)
+
+        try:
+            g._divcheck()
+        except NotInvertible:
+            raise ZeroDivisionError(f"{f} % {g}")
+
+        # Division where defined is always exact so there is no remainder
+        return f.ext.zero
+
+    def __rmod__(f, g):
+        try:
+            g = f.ext.convert(g)
+        except CoercionFailed:
+            return NotImplemented
+        return g % f
+
+    def __pow__(f, n):
+        if not isinstance(n, int):
+            raise TypeError("exponent of type 'int' expected")
+        if n < 0:
+            try:
+                f, n = f.inverse(), -n
+            except NotImplementedError:
+                raise ValueError("negative powers are not defined")
+
+        b = f.rep
+        m = f.ext.mod
+        r = f.ext.one.rep
+        while n > 0:
+            if n % 2:
+                r = (r*b) % m
+            b = (b*b) % m
+            n //= 2
+
+        return ExtElem(r, f.ext)
+
+    def __eq__(f, g):
+        if isinstance(g, ExtElem):
+            return f.rep == g.rep and f.ext == g.ext
+        else:
+            return NotImplemented
+
+    def __ne__(f, g):
+        return not f == g
+
+    def __hash__(f):
+        return hash((f.rep, f.ext))
+
+    def __str__(f):
+        from sympy.printing.str import sstr
+        return sstr(f.as_expr())
+
+    __repr__ = __str__
+
+    @property
+    def is_ground(f):
+        return f.rep.is_ground
+
+    def to_ground(f):
+        [c] = f.rep.to_list()
+        return c
+
+ExtElem = ExtensionElement
+
+
+class MonogenicFiniteExtension(Domain):
+    r"""
+    Finite extension generated by an integral element.
+
+    The generator is defined by a monic univariate
+    polynomial derived from the argument ``mod``.
+
+    A shorter alias is ``FiniteExtension``.
+
+    Examples
+    ========
+
+    Quadratic integer ring $\mathbb{Z}[\sqrt2]$:
+
+    >>> from sympy import Symbol, Poly
+    >>> from sympy.polys.agca.extensions import FiniteExtension
+    >>> x = Symbol('x')
+    >>> R = FiniteExtension(Poly(x**2 - 2)); R
+    ZZ[x]/(x**2 - 2)
+    >>> R.rank
+    2
+    >>> R(1 + x)*(3 - 2*x)
+    x - 1
+
+    Finite field $GF(5^3)$ defined by the primitive
+    polynomial $x^3 + x^2 + 2$ (over $\mathbb{Z}_5$).
+
+    >>> F = FiniteExtension(Poly(x**3 + x**2 + 2, modulus=5)); F
+    GF(5)[x]/(x**3 + x**2 + 2)
+    >>> F.basis
+    (1, x, x**2)
+    >>> F(x + 3)/(x**2 + 2)
+    -2*x**2 + x + 2
+
+    Function field of an elliptic curve:
+
+    >>> t = Symbol('t')
+    >>> FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
+    ZZ(x)[t]/(t**2 - x**3 - x + 1)
+
+    """
+    is_FiniteExtension = True
+
+    dtype = ExtensionElement
+
+    def __init__(self, mod):
+        if not (isinstance(mod, Poly) and mod.is_univariate):
+            raise TypeError("modulus must be a univariate Poly")
+
+        # Using auto=True (default) potentially changes the ground domain to a
+        # field whereas auto=False raises if division is not exact.  We'll let
+        # the caller decide whether or not they want to put the ground domain
+        # over a field. In most uses mod is already monic.
+        mod = mod.monic(auto=False)
+
+        self.rank = mod.degree()
+        self.modulus = mod
+        self.mod = mod.rep  # DMP representation
+
+        self.domain = dom = mod.domain
+        self.ring = dom.old_poly_ring(*mod.gens)
+
+        self.zero = self.convert(self.ring.zero)
+        self.one = self.convert(self.ring.one)
+
+        gen = self.ring.gens[0]
+        self.symbol = self.ring.symbols[0]
+        self.generator = self.convert(gen)
+        self.basis = tuple(self.convert(gen**i) for i in range(self.rank))
+
+        # XXX: It might be necessary to check mod.is_irreducible here
+        self.is_Field = self.domain.is_Field
+
+    def new(self, arg):
+        rep = self.ring.convert(arg)
+        return ExtElem(rep % self.mod, self)
+
+    def __eq__(self, other):
+        if not isinstance(other, FiniteExtension):
+            return False
+        return self.modulus == other.modulus
+
+    def __hash__(self):
+        return hash((self.__class__.__name__, self.modulus))
+
+    def __str__(self):
+        return "%s/(%s)" % (self.ring, self.modulus.as_expr())
+
+    __repr__ = __str__
+
+    @property
+    def has_CharacteristicZero(self):
+        return self.domain.has_CharacteristicZero
+
+    def characteristic(self):
+        return self.domain.characteristic()
+
+    def convert(self, f, base=None):
+        rep = self.ring.convert(f, base)
+        return ExtElem(rep % self.mod, self)
+
+    def convert_from(self, f, base):
+        rep = self.ring.convert(f, base)
+        return ExtElem(rep % self.mod, self)
+
+    def to_sympy(self, f):
+        return self.ring.to_sympy(f.rep)
+
+    def from_sympy(self, f):
+        return self.convert(f)
+
+    def set_domain(self, K):
+        mod = self.modulus.set_domain(K)
+        return self.__class__(mod)
+
+    def drop(self, *symbols):
+        if self.symbol in symbols:
+            raise GeneratorsError('Can not drop generator from FiniteExtension')
+        K = self.domain.drop(*symbols)
+        return self.set_domain(K)
+
+    def quo(self, f, g):
+        return self.exquo(f, g)
+
+    def exquo(self, f, g):
+        rep = self.ring.exquo(f.rep, g.rep)
+        return ExtElem(rep % self.mod, self)
+
+    def is_negative(self, a):
+        return False
+
+    def is_unit(self, a):
+        if self.is_Field:
+            return bool(a)
+        elif a.is_ground:
+            return self.domain.is_unit(a.to_ground())
+
+FiniteExtension = MonogenicFiniteExtension

rl/Lib/site-packages/sympy/polys/agca/homomorphisms.py

@@ -0,0 +1,691 @@
+"""
+Computations with homomorphisms of modules and rings.
+
+This module implements classes for representing homomorphisms of rings and
+their modules. Instead of instantiating the classes directly, you should use
+the function ``homomorphism(from, to, matrix)`` to create homomorphism objects.
+"""
+
+
+from sympy.polys.agca.modules import (Module, FreeModule, QuotientModule,
+    SubModule, SubQuotientModule)
+from sympy.polys.polyerrors import CoercionFailed
+
+# The main computational task for module homomorphisms is kernels.
+# For this reason, the concrete classes are organised by domain module type.
+
+
+class ModuleHomomorphism:
+    """
+    Abstract base class for module homomoprhisms. Do not instantiate.
+
+    Instead, use the ``homomorphism`` function:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> F = QQ.old_poly_ring(x).free_module(2)
+    >>> homomorphism(F, F, [[1, 0], [0, 1]])
+    Matrix([
+    [1, 0], : QQ[x]**2 -> QQ[x]**2
+    [0, 1]])
+
+    Attributes:
+
+    - ring - the ring over which we are considering modules
+    - domain - the domain module
+    - codomain - the codomain module
+    - _ker - cached kernel
+    - _img - cached image
+
+    Non-implemented methods:
+
+    - _kernel
+    - _image
+    - _restrict_domain
+    - _restrict_codomain
+    - _quotient_domain
+    - _quotient_codomain
+    - _apply
+    - _mul_scalar
+    - _compose
+    - _add
+    """
+
+    def __init__(self, domain, codomain):
+        if not isinstance(domain, Module):
+            raise TypeError('Source must be a module, got %s' % domain)
+        if not isinstance(codomain, Module):
+            raise TypeError('Target must be a module, got %s' % codomain)
+        if domain.ring != codomain.ring:
+            raise ValueError('Source and codomain must be over same ring, '
+                             'got %s != %s' % (domain, codomain))
+        self.domain = domain
+        self.codomain = codomain
+        self.ring = domain.ring
+        self._ker = None
+        self._img = None
+
+    def kernel(self):
+        r"""
+        Compute the kernel of ``self``.
+
+        That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute
+        `ker(\phi) = \{x \in M | \phi(x) = 0\}`.  This is a submodule of `M`.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> homomorphism(F, F, [[1, 0], [x, 0]]).kernel()
+        <[x, -1]>
+        """
+        if self._ker is None:
+            self._ker = self._kernel()
+        return self._ker
+
+    def image(self):
+        r"""
+        Compute the image of ``self``.
+
+        That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute
+        `im(\phi) = \{\phi(x) | x \in M \}`.  This is a submodule of `N`.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> homomorphism(F, F, [[1, 0], [x, 0]]).image() == F.submodule([1, 0])
+        True
+        """
+        if self._img is None:
+            self._img = self._image()
+        return self._img
+
+    def _kernel(self):
+        """Compute the kernel of ``self``."""
+        raise NotImplementedError
+
+    def _image(self):
+        """Compute the image of ``self``."""
+        raise NotImplementedError
+
+    def _restrict_domain(self, sm):
+        """Implementation of domain restriction."""
+        raise NotImplementedError
+
+    def _restrict_codomain(self, sm):
+        """Implementation of codomain restriction."""
+        raise NotImplementedError
+
+    def _quotient_domain(self, sm):
+        """Implementation of domain quotient."""
+        raise NotImplementedError
+
+    def _quotient_codomain(self, sm):
+        """Implementation of codomain quotient."""
+        raise NotImplementedError
+
+    def restrict_domain(self, sm):
+        """
+        Return ``self``, with the domain restricted to ``sm``.
+
+        Here ``sm`` has to be a submodule of ``self.domain``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.restrict_domain(F.submodule([1, 0]))
+        Matrix([
+        [1, x], : <[1, 0]> -> QQ[x]**2
+        [0, 0]])
+
+        This is the same as just composing on the right with the submodule
+        inclusion:
+
+        >>> h * F.submodule([1, 0]).inclusion_hom()
+        Matrix([
+        [1, x], : <[1, 0]> -> QQ[x]**2
+        [0, 0]])
+        """
+        if not self.domain.is_submodule(sm):
+            raise ValueError('sm must be a submodule of %s, got %s'
+                             % (self.domain, sm))
+        if sm == self.domain:
+            return self
+        return self._restrict_domain(sm)
+
+    def restrict_codomain(self, sm):
+        """
+        Return ``self``, with codomain restricted to to ``sm``.
+
+        Here ``sm`` has to be a submodule of ``self.codomain`` containing the
+        image.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.restrict_codomain(F.submodule([1, 0]))
+        Matrix([
+        [1, x], : QQ[x]**2 -> <[1, 0]>
+        [0, 0]])
+        """
+        if not sm.is_submodule(self.image()):
+            raise ValueError('the image %s must contain sm, got %s'
+                             % (self.image(), sm))
+        if sm == self.codomain:
+            return self
+        return self._restrict_codomain(sm)
+
+    def quotient_domain(self, sm):
+        """
+        Return ``self`` with domain replaced by ``domain/sm``.
+
+        Here ``sm`` must be a submodule of ``self.kernel()``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.quotient_domain(F.submodule([-x, 1]))
+        Matrix([
+        [1, x], : QQ[x]**2/<[-x, 1]> -> QQ[x]**2
+        [0, 0]])
+        """
+        if not self.kernel().is_submodule(sm):
+            raise ValueError('kernel %s must contain sm, got %s' %
+                             (self.kernel(), sm))
+        if sm.is_zero():
+            return self
+        return self._quotient_domain(sm)
+
+    def quotient_codomain(self, sm):
+        """
+        Return ``self`` with codomain replaced by ``codomain/sm``.
+
+        Here ``sm`` must be a submodule of ``self.codomain``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2
+        [0, 0]])
+        >>> h.quotient_codomain(F.submodule([1, 1]))
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2/<[1, 1]>
+        [0, 0]])
+
+        This is the same as composing with the quotient map on the left:
+
+        >>> (F/[(1, 1)]).quotient_hom() * h
+        Matrix([
+        [1, x], : QQ[x]**2 -> QQ[x]**2/<[1, 1]>
+        [0, 0]])
+        """
+        if not self.codomain.is_submodule(sm):
+            raise ValueError('sm must be a submodule of codomain %s, got %s'
+                             % (self.codomain, sm))
+        if sm.is_zero():
+            return self
+        return self._quotient_codomain(sm)
+
+    def _apply(self, elem):
+        """Apply ``self`` to ``elem``."""
+        raise NotImplementedError
+
+    def __call__(self, elem):
+        return self.codomain.convert(self._apply(self.domain.convert(elem)))
+
+    def _compose(self, oth):
+        """
+        Compose ``self`` with ``oth``, that is, return the homomorphism
+        obtained by first applying then ``self``, then ``oth``.
+
+        (This method is private since in this syntax, it is non-obvious which
+        homomorphism is executed first.)
+        """
+        raise NotImplementedError
+
+    def _mul_scalar(self, c):
+        """Scalar multiplication. ``c`` is guaranteed in self.ring."""
+        raise NotImplementedError
+
+    def _add(self, oth):
+        """
+        Homomorphism addition.
+        ``oth`` is guaranteed to be a homomorphism with same domain/codomain.
+        """
+        raise NotImplementedError
+
+    def _check_hom(self, oth):
+        """Helper to check that oth is a homomorphism with same domain/codomain."""
+        if not isinstance(oth, ModuleHomomorphism):
+            return False
+        return oth.domain == self.domain and oth.codomain == self.codomain
+
+    def __mul__(self, oth):
+        if isinstance(oth, ModuleHomomorphism) and self.domain == oth.codomain:
+            return oth._compose(self)
+        try:
+            return self._mul_scalar(self.ring.convert(oth))
+        except CoercionFailed:
+            return NotImplemented
+
+    # NOTE: _compose will never be called from rmul
+    __rmul__ = __mul__
+
+    def __truediv__(self, oth):
+        try:
+            return self._mul_scalar(1/self.ring.convert(oth))
+        except CoercionFailed:
+            return NotImplemented
+
+    def __add__(self, oth):
+        if self._check_hom(oth):
+            return self._add(oth)
+        return NotImplemented
+
+    def __sub__(self, oth):
+        if self._check_hom(oth):
+            return self._add(oth._mul_scalar(self.ring.convert(-1)))
+        return NotImplemented
+
+    def is_injective(self):
+        """
+        Return True if ``self`` is injective.
+
+        That is, check if the elements of the domain are mapped to the same
+        codomain element.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h.is_injective()
+        False
+        >>> h.quotient_domain(h.kernel()).is_injective()
+        True
+        """
+        return self.kernel().is_zero()
+
+    def is_surjective(self):
+        """
+        Return True if ``self`` is surjective.
+
+        That is, check if every element of the codomain has at least one
+        preimage.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h.is_surjective()
+        False
+        >>> h.restrict_codomain(h.image()).is_surjective()
+        True
+        """
+        return self.image() == self.codomain
+
+    def is_isomorphism(self):
+        """
+        Return True if ``self`` is an isomorphism.
+
+        That is, check if every element of the codomain has precisely one
+        preimage. Equivalently, ``self`` is both injective and surjective.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h = h.restrict_codomain(h.image())
+        >>> h.is_isomorphism()
+        False
+        >>> h.quotient_domain(h.kernel()).is_isomorphism()
+        True
+        """
+        return self.is_injective() and self.is_surjective()
+
+    def is_zero(self):
+        """
+        Return True if ``self`` is a zero morphism.
+
+        That is, check if every element of the domain is mapped to zero
+        under self.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x
+        >>> from sympy.polys.agca import homomorphism
+
+        >>> F = QQ.old_poly_ring(x).free_module(2)
+        >>> h = homomorphism(F, F, [[1, 0], [x, 0]])
+        >>> h.is_zero()
+        False
+        >>> h.restrict_domain(F.submodule()).is_zero()
+        True
+        >>> h.quotient_codomain(h.image()).is_zero()
+        True
+        """
+        return self.image().is_zero()
+
+    def __eq__(self, oth):
+        try:
+            return (self - oth).is_zero()
+        except TypeError:
+            return False
+
+    def __ne__(self, oth):
+        return not (self == oth)
+
+
+class MatrixHomomorphism(ModuleHomomorphism):
+    r"""
+    Helper class for all homomoprhisms which are expressed via a matrix.
+
+    That is, for such homomorphisms ``domain`` is contained in a module
+    generated by finitely many elements `e_1, \ldots, e_n`, so that the
+    homomorphism is determined uniquely by its action on the `e_i`. It
+    can thus be represented as a vector of elements of the codomain module,
+    or potentially a supermodule of the codomain module
+    (and hence conventionally as a matrix, if there is a similar interpretation
+    for elements of the codomain module).
+
+    Note that this class does *not* assume that the `e_i` freely generate a
+    submodule, nor that ``domain`` is even all of this submodule. It exists
+    only to unify the interface.
+
+    Do not instantiate.
+
+    Attributes:
+
+    - matrix - the list of images determining the homomorphism.
+    NOTE: the elements of matrix belong to either self.codomain or
+          self.codomain.container
+
+    Still non-implemented methods:
+
+    - kernel
+    - _apply
+    """
+
+    def __init__(self, domain, codomain, matrix):
+        ModuleHomomorphism.__init__(self, domain, codomain)
+        if len(matrix) != domain.rank:
+            raise ValueError('Need to provide %s elements, got %s'
+                             % (domain.rank, len(matrix)))
+
+        converter = self.codomain.convert
+        if isinstance(self.codomain, (SubModule, SubQuotientModule)):
+            converter = self.codomain.container.convert
+        self.matrix = tuple(converter(x) for x in matrix)
+
+    def _sympy_matrix(self):
+        """Helper function which returns a SymPy matrix ``self.matrix``."""
+        from sympy.matrices import Matrix
+        c = lambda x: x
+        if isinstance(self.codomain, (QuotientModule, SubQuotientModule)):
+            c = lambda x: x.data
+        return Matrix([[self.ring.to_sympy(y) for y in c(x)] for x in self.matrix]).T
+
+    def __repr__(self):
+        lines = repr(self._sympy_matrix()).split('\n')
+        t = " : %s -> %s" % (self.domain, self.codomain)
+        s = ' '*len(t)
+        n = len(lines)
+        for i in range(n // 2):
+            lines[i] += s
+        lines[n // 2] += t
+        for i in range(n//2 + 1, n):
+            lines[i] += s
+        return '\n'.join(lines)
+
+    def _restrict_domain(self, sm):
+        """Implementation of domain restriction."""
+        return SubModuleHomomorphism(sm, self.codomain, self.matrix)
+
+    def _restrict_codomain(self, sm):
+        """Implementation of codomain restriction."""
+        return self.__class__(self.domain, sm, self.matrix)
+
+    def _quotient_domain(self, sm):
+        """Implementation of domain quotient."""
+        return self.__class__(self.domain/sm, self.codomain, self.matrix)
+
+    def _quotient_codomain(self, sm):
+        """Implementation of codomain quotient."""
+        Q = self.codomain/sm
+        converter = Q.convert
+        if isinstance(self.codomain, SubModule):
+            converter = Q.container.convert
+        return self.__class__(self.domain, self.codomain/sm,
+            [converter(x) for x in self.matrix])
+
+    def _add(self, oth):
+        return self.__class__(self.domain, self.codomain,
+                              [x + y for x, y in zip(self.matrix, oth.matrix)])
+
+    def _mul_scalar(self, c):
+        return self.__class__(self.domain, self.codomain, [c*x for x in self.matrix])
+
+    def _compose(self, oth):
+        return self.__class__(self.domain, oth.codomain, [oth(x) for x in self.matrix])
+
+
+class FreeModuleHomomorphism(MatrixHomomorphism):
+    """
+    Concrete class for homomorphisms with domain a free module or a quotient
+    thereof.
+
+    Do not instantiate; the constructor does not check that your data is well
+    defined. Use the ``homomorphism`` function instead:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> F = QQ.old_poly_ring(x).free_module(2)
+    >>> homomorphism(F, F, [[1, 0], [0, 1]])
+    Matrix([
+    [1, 0], : QQ[x]**2 -> QQ[x]**2
+    [0, 1]])
+    """
+
+    def _apply(self, elem):
+        if isinstance(self.domain, QuotientModule):
+            elem = elem.data
+        return sum(x * e for x, e in zip(elem, self.matrix))
+
+    def _image(self):
+        return self.codomain.submodule(*self.matrix)
+
+    def _kernel(self):
+        # The domain is either a free module or a quotient thereof.
+        # It does not matter if it is a quotient, because that won't increase
+        # the kernel.
+        # Our generators {e_i} are sent to the matrix entries {b_i}.
+        # The kernel is essentially the syzygy module of these {b_i}.
+        syz = self.image().syzygy_module()
+        return self.domain.submodule(*syz.gens)
+
+
+class SubModuleHomomorphism(MatrixHomomorphism):
+    """
+    Concrete class for homomorphism with domain a submodule of a free module
+    or a quotient thereof.
+
+    Do not instantiate; the constructor does not check that your data is well
+    defined. Use the ``homomorphism`` function instead:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> M = QQ.old_poly_ring(x).free_module(2)*x
+    >>> homomorphism(M, M, [[1, 0], [0, 1]])
+    Matrix([
+    [1, 0], : <[x, 0], [0, x]> -> <[x, 0], [0, x]>
+    [0, 1]])
+    """
+
+    def _apply(self, elem):
+        if isinstance(self.domain, SubQuotientModule):
+            elem = elem.data
+        return sum(x * e for x, e in zip(elem, self.matrix))
+
+    def _image(self):
+        return self.codomain.submodule(*[self(x) for x in self.domain.gens])
+
+    def _kernel(self):
+        syz = self.image().syzygy_module()
+        return self.domain.submodule(
+            *[sum(xi*gi for xi, gi in zip(s, self.domain.gens))
+              for s in syz.gens])
+
+
+def homomorphism(domain, codomain, matrix):
+    r"""
+    Create a homomorphism object.
+
+    This function tries to build a homomorphism from ``domain`` to ``codomain``
+    via the matrix ``matrix``.
+
+    Examples
+    ========
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> from sympy.polys.agca import homomorphism
+
+    >>> R = QQ.old_poly_ring(x)
+    >>> T = R.free_module(2)
+
+    If ``domain`` is a free module generated by `e_1, \ldots, e_n`, then
+    ``matrix`` should be an n-element iterable `(b_1, \ldots, b_n)` where
+    the `b_i` are elements of ``codomain``. The constructed homomorphism is the
+    unique homomorphism sending `e_i` to `b_i`.
+
+    >>> F = R.free_module(2)
+    >>> h = homomorphism(F, T, [[1, x], [x**2, 0]])
+    >>> h
+    Matrix([
+    [1, x**2], : QQ[x]**2 -> QQ[x]**2
+    [x,    0]])
+    >>> h([1, 0])
+    [1, x]
+    >>> h([0, 1])
+    [x**2, 0]
+    >>> h([1, 1])
+    [x**2 + 1, x]
+
+    If ``domain`` is a submodule of a free module, them ``matrix`` determines
+    a homomoprhism from the containing free module to ``codomain``, and the
+    homomorphism returned is obtained by restriction to ``domain``.
+
+    >>> S = F.submodule([1, 0], [0, x])
+    >>> homomorphism(S, T, [[1, x], [x**2, 0]])
+    Matrix([
+    [1, x**2], : <[1, 0], [0, x]> -> QQ[x]**2
+    [x,    0]])
+
+    If ``domain`` is a (sub)quotient `N/K`, then ``matrix`` determines a
+    homomorphism from `N` to ``codomain``. If the kernel contains `K`, this
+    homomorphism descends to ``domain`` and is returned; otherwise an exception
+    is raised.
+
+    >>> homomorphism(S/[(1, 0)], T, [0, [x**2, 0]])
+    Matrix([
+    [0, x**2], : <[1, 0] + <[1, 0]>, [0, x] + <[1, 0]>, [1, 0] + <[1, 0]>> -> QQ[x]**2
+    [0,    0]])
+    >>> homomorphism(S/[(0, x)], T, [0, [x**2, 0]])
+    Traceback (most recent call last):
+    ...
+    ValueError: kernel <[1, 0], [0, 0]> must contain sm, got <[0,x]>
+
+    """
+    def freepres(module):
+        """
+        Return a tuple ``(F, S, Q, c)`` where ``F`` is a free module, ``S`` is a
+        submodule of ``F``, and ``Q`` a submodule of ``S``, such that
+        ``module = S/Q``, and ``c`` is a conversion function.
+        """
+        if isinstance(module, FreeModule):
+            return module, module, module.submodule(), lambda x: module.convert(x)
+        if isinstance(module, QuotientModule):
+            return (module.base, module.base, module.killed_module,
+                    lambda x: module.convert(x).data)
+        if isinstance(module, SubQuotientModule):
+            return (module.base.container, module.base, module.killed_module,
+                    lambda x: module.container.convert(x).data)
+        # an ordinary submodule
+        return (module.container, module, module.submodule(),
+                lambda x: module.container.convert(x))
+
+    SF, SS, SQ, _ = freepres(domain)
+    TF, TS, TQ, c = freepres(codomain)
+    # NOTE this is probably a bit inefficient (redundant checks)
+    return FreeModuleHomomorphism(SF, TF, [c(x) for x in matrix]
+         ).restrict_domain(SS).restrict_codomain(TS
+         ).quotient_codomain(TQ).quotient_domain(SQ)

rl/Lib/site-packages/sympy/polys/agca/ideals.py

@@ -0,0 +1,395 @@
+"""Computations with ideals of polynomial rings."""
+
+from sympy.polys.polyerrors import CoercionFailed
+from sympy.polys.polyutils import IntegerPowerable
+
+
+class Ideal(IntegerPowerable):
+    """
+    Abstract base class for ideals.
+
+    Do not instantiate - use explicit constructors in the ring class instead:
+
+    >>> from sympy import QQ
+    >>> from sympy.abc import x
+    >>> QQ.old_poly_ring(x).ideal(x+1)
+    <x + 1>
+
+    Attributes
+
+    - ring - the ring this ideal belongs to
+
+    Non-implemented methods:
+
+    - _contains_elem
+    - _contains_ideal
+    - _quotient
+    - _intersect
+    - _union
+    - _product
+    - is_whole_ring
+    - is_zero
+    - is_prime, is_maximal, is_primary, is_radical
+    - is_principal
+    - height, depth
+    - radical
+
+    Methods that likely should be overridden in subclasses:
+
+    - reduce_element
+    """
+
+    def _contains_elem(self, x):
+        """Implementation of element containment."""
+        raise NotImplementedError
+
+    def _contains_ideal(self, I):
+        """Implementation of ideal containment."""
+        raise NotImplementedError
+
+    def _quotient(self, J):
+        """Implementation of ideal quotient."""
+        raise NotImplementedError
+
+    def _intersect(self, J):
+        """Implementation of ideal intersection."""
+        raise NotImplementedError
+
+    def is_whole_ring(self):
+        """Return True if ``self`` is the whole ring."""
+        raise NotImplementedError
+
+    def is_zero(self):
+        """Return True if ``self`` is the zero ideal."""
+        raise NotImplementedError
+
+    def _equals(self, J):
+        """Implementation of ideal equality."""
+        return self._contains_ideal(J) and J._contains_ideal(self)
+
+    def is_prime(self):
+        """Return True if ``self`` is a prime ideal."""
+        raise NotImplementedError
+
+    def is_maximal(self):
+        """Return True if ``self`` is a maximal ideal."""
+        raise NotImplementedError
+
+    def is_radical(self):
+        """Return True if ``self`` is a radical ideal."""
+        raise NotImplementedError
+
+    def is_primary(self):
+        """Return True if ``self`` is a primary ideal."""
+        raise NotImplementedError
+
+    def is_principal(self):
+        """Return True if ``self`` is a principal ideal."""
+        raise NotImplementedError
+
+    def radical(self):
+        """Compute the radical of ``self``."""
+        raise NotImplementedError
+
+    def depth(self):
+        """Compute the depth of ``self``."""
+        raise NotImplementedError
+
+    def height(self):
+        """Compute the height of ``self``."""
+        raise NotImplementedError
+
+    # TODO more
+
+    # non-implemented methods end here
+
+    def __init__(self, ring):
+        self.ring = ring
+
+    def _check_ideal(self, J):
+        """Helper to check ``J`` is an ideal of our ring."""
+        if not isinstance(J, Ideal) or J.ring != self.ring:
+            raise ValueError(
+                'J must be an ideal of %s, got %s' % (self.ring, J))
+
+    def contains(self, elem):
+        """
+        Return True if ``elem`` is an element of this ideal.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x+1, x-1).contains(3)
+        True
+        >>> QQ.old_poly_ring(x).ideal(x**2, x**3).contains(x)
+        False
+        """
+        return self._contains_elem(self.ring.convert(elem))
+
+    def subset(self, other):
+        """
+        Returns True if ``other`` is is a subset of ``self``.
+
+        Here ``other`` may be an ideal.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> I = QQ.old_poly_ring(x).ideal(x+1)
+        >>> I.subset([x**2 - 1, x**2 + 2*x + 1])
+        True
+        >>> I.subset([x**2 + 1, x + 1])
+        False
+        >>> I.subset(QQ.old_poly_ring(x).ideal(x**2 - 1))
+        True
+        """
+        if isinstance(other, Ideal):
+            return self._contains_ideal(other)
+        return all(self._contains_elem(x) for x in other)
+
+    def quotient(self, J, **opts):
+        r"""
+        Compute the ideal quotient of ``self`` by ``J``.
+
+        That is, if ``self`` is the ideal `I`, compute the set
+        `I : J = \{x \in R | xJ \subset I \}`.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import QQ
+        >>> R = QQ.old_poly_ring(x, y)
+        >>> R.ideal(x*y).quotient(R.ideal(x))
+        <y>
+        """
+        self._check_ideal(J)
+        return self._quotient(J, **opts)
+
+    def intersect(self, J):
+        """
+        Compute the intersection of self with ideal J.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import QQ
+        >>> R = QQ.old_poly_ring(x, y)
+        >>> R.ideal(x).intersect(R.ideal(y))
+        <x*y>
+        """
+        self._check_ideal(J)
+        return self._intersect(J)
+
+    def saturate(self, J):
+        r"""
+        Compute the ideal saturation of ``self`` by ``J``.
+
+        That is, if ``self`` is the ideal `I`, compute the set
+        `I : J^\infty = \{x \in R | xJ^n \subset I \text{ for some } n\}`.
+        """
+        raise NotImplementedError
+        # Note this can be implemented using repeated quotient
+
+    def union(self, J):
+        """
+        Compute the ideal generated by the union of ``self`` and ``J``.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x**2 - 1).union(QQ.old_poly_ring(x).ideal((x+1)**2)) == QQ.old_poly_ring(x).ideal(x+1)
+        True
+        """
+        self._check_ideal(J)
+        return self._union(J)
+
+    def product(self, J):
+        r"""
+        Compute the ideal product of ``self`` and ``J``.
+
+        That is, compute the ideal generated by products `xy`, for `x` an element
+        of ``self`` and `y \in J`.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x, y
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x, y).ideal(x).product(QQ.old_poly_ring(x, y).ideal(y))
+        <x*y>
+        """
+        self._check_ideal(J)
+        return self._product(J)
+
+    def reduce_element(self, x):
+        """
+        Reduce the element ``x`` of our ring modulo the ideal ``self``.
+
+        Here "reduce" has no specific meaning: it could return a unique normal
+        form, simplify the expression a bit, or just do nothing.
+        """
+        return x
+
+    def __add__(self, e):
+        if not isinstance(e, Ideal):
+            R = self.ring.quotient_ring(self)
+            if isinstance(e, R.dtype):
+                return e
+            if isinstance(e, R.ring.dtype):
+                return R(e)
+            return R.convert(e)
+        self._check_ideal(e)
+        return self.union(e)
+
+    __radd__ = __add__
+
+    def __mul__(self, e):
+        if not isinstance(e, Ideal):
+            try:
+                e = self.ring.ideal(e)
+            except CoercionFailed:
+                return NotImplemented
+        self._check_ideal(e)
+        return self.product(e)
+
+    __rmul__ = __mul__
+
+    def _zeroth_power(self):
+        return self.ring.ideal(1)
+
+    def _first_power(self):
+        # Raising to any power but 1 returns a new instance. So we mult by 1
+        # here so that the first power is no exception.
+        return self * 1
+
+    def __eq__(self, e):
+        if not isinstance(e, Ideal) or e.ring != self.ring:
+            return False
+        return self._equals(e)
+
+    def __ne__(self, e):
+        return not (self == e)
+
+
+class ModuleImplementedIdeal(Ideal):
+    """
+    Ideal implementation relying on the modules code.
+
+    Attributes:
+
+    - _module - the underlying module
+    """
+
+    def __init__(self, ring, module):
+        Ideal.__init__(self, ring)
+        self._module = module
+
+    def _contains_elem(self, x):
+        return self._module.contains([x])
+
+    def _contains_ideal(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self._module.is_submodule(J._module)
+
+    def _intersect(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self.__class__(self.ring, self._module.intersect(J._module))
+
+    def _quotient(self, J, **opts):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self._module.module_quotient(J._module, **opts)
+
+    def _union(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self.__class__(self.ring, self._module.union(J._module))
+
+    @property
+    def gens(self):
+        """
+        Return generators for ``self``.
+
+        Examples
+        ========
+
+        >>> from sympy import QQ
+        >>> from sympy.abc import x, y
+        >>> list(QQ.old_poly_ring(x, y).ideal(x, y, x**2 + y).gens)
+        [DMP_Python([[1], []], QQ), DMP_Python([[1, 0]], QQ), DMP_Python([[1], [], [1, 0]], QQ)]
+        """
+        return (x[0] for x in self._module.gens)
+
+    def is_zero(self):
+        """
+        Return True if ``self`` is the zero ideal.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> QQ.old_poly_ring(x).ideal(x).is_zero()
+        False
+        >>> QQ.old_poly_ring(x).ideal().is_zero()
+        True
+        """
+        return self._module.is_zero()
+
+    def is_whole_ring(self):
+        """
+        Return True if ``self`` is the whole ring, i.e. one generator is a unit.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ, ilex
+        >>> QQ.old_poly_ring(x).ideal(x).is_whole_ring()
+        False
+        >>> QQ.old_poly_ring(x).ideal(3).is_whole_ring()
+        True
+        >>> QQ.old_poly_ring(x, order=ilex).ideal(2 + x).is_whole_ring()
+        True
+        """
+        return self._module.is_full_module()
+
+    def __repr__(self):
+        from sympy.printing.str import sstr
+        gens = [self.ring.to_sympy(x) for [x] in self._module.gens]
+        return '<' + ','.join(sstr(g) for g in gens) + '>'
+
+    # NOTE this is the only method using the fact that the module is a SubModule
+    def _product(self, J):
+        if not isinstance(J, ModuleImplementedIdeal):
+            raise NotImplementedError
+        return self.__class__(self.ring, self._module.submodule(
+            *[[x*y] for [x] in self._module.gens for [y] in J._module.gens]))
+
+    def in_terms_of_generators(self, e):
+        """
+        Express ``e`` in terms of the generators of ``self``.
+
+        Examples
+        ========
+
+        >>> from sympy.abc import x
+        >>> from sympy import QQ
+        >>> I = QQ.old_poly_ring(x).ideal(x**2 + 1, x)
+        >>> I.in_terms_of_generators(1)  # doctest: +SKIP
+        [DMP_Python([1], QQ), DMP_Python([-1, 0], QQ)]
+        """
+        return self._module.in_terms_of_generators([e])
+
+    def reduce_element(self, x, **options):
+        return self._module.reduce_element([x], **options)[0]

rl/Lib/site-packages/sympy/polys/agca/tests/test_extensions.py

@@ -0,0 +1,196 @@
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.polys import QQ, ZZ
+from sympy.polys.polytools import Poly
+from sympy.polys.polyerrors import NotInvertible
+from sympy.polys.agca.extensions import FiniteExtension
+from sympy.polys.domainmatrix import DomainMatrix
+
+from sympy.testing.pytest import raises
+
+from sympy.abc import x, y, t
+
+
+def test_FiniteExtension():
+    # Gaussian integers
+    A = FiniteExtension(Poly(x**2 + 1, x))
+    assert A.rank == 2
+    assert str(A) == 'ZZ[x]/(x**2 + 1)'
+    i = A.generator
+    assert i.parent() is A
+
+    assert i*i == A(-1)
+    raises(TypeError, lambda: i*())
+
+    assert A.basis == (A.one, i)
+    assert A(1) == A.one
+    assert i**2 == A(-1)
+    assert i**2 != -1  # no coercion
+    assert (2 + i)*(1 - i) == 3 - i
+    assert (1 + i)**8 == A(16)
+    assert A(1).inverse() == A(1)
+    raises(NotImplementedError, lambda: A(2).inverse())
+
+    # Finite field of order 27
+    F = FiniteExtension(Poly(x**3 - x + 1, x, modulus=3))
+    assert F.rank == 3
+    a = F.generator  # also generates the cyclic group F - {0}
+    assert F.basis == (F(1), a, a**2)
+    assert a**27 == a
+    assert a**26 == F(1)
+    assert a**13 == F(-1)
+    assert a**9 == a + 1
+    assert a**3 == a - 1
+    assert a**6 == a**2 + a + 1
+    assert F(x**2 + x).inverse() == 1 - a
+    assert F(x + 2)**(-1) == F(x + 2).inverse()
+    assert a**19 * a**(-19) == F(1)
+    assert (a - 1) / (2*a**2 - 1) == a**2 + 1
+    assert (a - 1) // (2*a**2 - 1) == a**2 + 1
+    assert 2/(a**2 + 1) == a**2 - a + 1
+    assert (a**2 + 1)/2 == -a**2 - 1
+    raises(NotInvertible, lambda: F(0).inverse())
+
+    # Function field of an elliptic curve
+    K = FiniteExtension(Poly(t**2 - x**3 - x + 1, t, field=True))
+    assert K.rank == 2
+    assert str(K) == 'ZZ(x)[t]/(t**2 - x**3 - x + 1)'
+    y = K.generator
+    c = 1/(x**3 - x**2 + x - 1)
+    assert ((y + x)*(y - x)).inverse() == K(c)
+    assert (y + x)*(y - x)*c == K(1)  # explicit inverse of y + x
+
+
+def test_FiniteExtension_eq_hash():
+    # Test eq and hash
+    p1 = Poly(x**2 - 2, x, domain=ZZ)
+    p2 = Poly(x**2 - 2, x, domain=QQ)
+    K1 = FiniteExtension(p1)
+    K2 = FiniteExtension(p2)
+    assert K1 == FiniteExtension(Poly(x**2 - 2))
+    assert K2 != FiniteExtension(Poly(x**2 - 2))
+    assert len({K1, K2, FiniteExtension(p1)}) == 2
+
+
+def test_FiniteExtension_mod():
+    # Test mod
+    K = FiniteExtension(Poly(x**3 + 1, x, domain=QQ))
+    xf = K(x)
+    assert (xf**2 - 1) % 1 == K.zero
+    assert 1 % (xf**2 - 1) == K.zero
+    assert (xf**2 - 1) / (xf - 1) == xf + 1
+    assert (xf**2 - 1) // (xf - 1) == xf + 1
+    assert (xf**2 - 1) % (xf - 1) == K.zero
+    raises(ZeroDivisionError, lambda: (xf**2 - 1) % 0)
+    raises(TypeError, lambda: xf % [])
+    raises(TypeError, lambda: [] % xf)
+
+    # Test mod over ring
+    K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
+    xf = K(x)
+    assert (xf**2 - 1) % 1 == K.zero
+    raises(NotImplementedError, lambda: (xf**2 - 1) % (xf - 1))
+
+
+def test_FiniteExtension_from_sympy():
+    # Test to_sympy/from_sympy
+    K = FiniteExtension(Poly(x**3 + 1, x, domain=ZZ))
+    xf = K(x)
+    assert K.from_sympy(x) == xf
+    assert K.to_sympy(xf) == x
+
+
+def test_FiniteExtension_set_domain():
+    KZ = FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))
+    KQ = FiniteExtension(Poly(x**2 + 1, x, domain='QQ'))
+    assert KZ.set_domain(QQ) == KQ
+
+
+def test_FiniteExtension_exquo():
+    # Test exquo
+    K = FiniteExtension(Poly(x**4 + 1))
+    xf = K(x)
+    assert K.exquo(xf**2 - 1, xf - 1) == xf + 1
+
+
+def test_FiniteExtension_convert():
+    # Test from_MonogenicFiniteExtension
+    K1 = FiniteExtension(Poly(x**2 + 1))
+    K2 = QQ[x]
+    x1, x2 = K1(x), K2(x)
+    assert K1.convert(x2) == x1
+    assert K2.convert(x1) == x2
+
+    K = FiniteExtension(Poly(x**2 - 1, domain=QQ))
+    assert K.convert_from(QQ(1, 2), QQ) == K.one/2
+
+
+def test_FiniteExtension_division_ring():
+    # Test division in FiniteExtension over a ring
+    KQ = FiniteExtension(Poly(x**2 - 1, x, domain=QQ))
+    KZ = FiniteExtension(Poly(x**2 - 1, x, domain=ZZ))
+    KQt = FiniteExtension(Poly(x**2 - 1, x, domain=QQ[t]))
+    KQtf = FiniteExtension(Poly(x**2 - 1, x, domain=QQ.frac_field(t)))
+    assert KQ.is_Field is True
+    assert KZ.is_Field is False
+    assert KQt.is_Field is False
+    assert KQtf.is_Field is True
+    for K in KQ, KZ, KQt, KQtf:
+        xK = K.convert(x)
+        assert xK / K.one == xK
+        assert xK // K.one == xK
+        assert xK % K.one == K.zero
+        raises(ZeroDivisionError, lambda: xK / K.zero)
+        raises(ZeroDivisionError, lambda: xK // K.zero)
+        raises(ZeroDivisionError, lambda: xK % K.zero)
+        if K.is_Field:
+            assert xK / xK == K.one
+            assert xK // xK == K.one
+            assert xK % xK == K.zero
+        else:
+            raises(NotImplementedError, lambda: xK / xK)
+            raises(NotImplementedError, lambda: xK // xK)
+            raises(NotImplementedError, lambda: xK % xK)
+
+
+def test_FiniteExtension_Poly():
+    K = FiniteExtension(Poly(x**2 - 2))
+    p = Poly(x, y, domain=K)
+    assert p.domain == K
+    assert p.as_expr() == x
+    assert (p**2).as_expr() == 2
+
+    K = FiniteExtension(Poly(x**2 - 2, x, domain=QQ))
+    K2 = FiniteExtension(Poly(t**2 - 2, t, domain=K))
+    assert str(K2) == 'QQ[x]/(x**2 - 2)[t]/(t**2 - 2)'
+
+    eK = K2.convert(x + t)
+    assert K2.to_sympy(eK) == x + t
+    assert K2.to_sympy(eK ** 2) == 4 + 2*x*t
+    p = Poly(x + t, y, domain=K2)
+    assert p**2 == Poly(4 + 2*x*t, y, domain=K2)
+
+
+def test_FiniteExtension_sincos_jacobian():
+    # Use FiniteExtensino to compute the Jacobian of a matrix involving sin
+    # and cos of different symbols.
+    r, p, t = symbols('rho, phi, theta')
+    elements = [
+        [sin(p)*cos(t), r*cos(p)*cos(t), -r*sin(p)*sin(t)],
+        [sin(p)*sin(t), r*cos(p)*sin(t),  r*sin(p)*cos(t)],
+        [       cos(p),       -r*sin(p),                0],
+    ]
+
+    def make_extension(K):
+        K = FiniteExtension(Poly(sin(p)**2+cos(p)**2-1, sin(p), domain=K[cos(p)]))
+        K = FiniteExtension(Poly(sin(t)**2+cos(t)**2-1, sin(t), domain=K[cos(t)]))
+        return K
+
+    Ksc1 = make_extension(ZZ[r])
+    Ksc2 = make_extension(ZZ)[r]
+
+    for K in [Ksc1, Ksc2]:
+        elements_K = [[K.convert(e) for e in row] for row in elements]
+        J = DomainMatrix(elements_K, (3, 3), K)
+        det = J.charpoly()[-1] * (-K.one)**3
+        assert det == K.convert(r**2*sin(p))

rl/Lib/site-packages/sympy/polys/agca/tests/test_homomorphisms.py

@@ -0,0 +1,113 @@
+"""Tests for homomorphisms."""
+
+from sympy.core.singleton import S
+from sympy.polys.domains.rationalfield import QQ
+from sympy.abc import x, y
+from sympy.polys.agca import homomorphism
+from sympy.testing.pytest import raises
+
+
+def test_printing():
+    R = QQ.old_poly_ring(x)
+
+    assert str(homomorphism(R.free_module(1), R.free_module(1), [0])) == \
+        'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1'
+    assert str(homomorphism(R.free_module(2), R.free_module(2), [0, 0])) == \
+        'Matrix([                       \n[0, 0], : QQ[x]**2 -> QQ[x]**2\n[0, 0]])                       '
+    assert str(homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])) == \
+        'Matrix([[0]]) : QQ[x]**1 -> QQ[x]**1/<[x]>'
+    assert str(R.free_module(0).identity_hom()) == 'Matrix(0, 0, []) : QQ[x]**0 -> QQ[x]**0'
+
+def test_operations():
+    F = QQ.old_poly_ring(x).free_module(2)
+    G = QQ.old_poly_ring(x).free_module(3)
+    f = F.identity_hom()
+    g = homomorphism(F, F, [0, [1, x]])
+    h = homomorphism(F, F, [[1, 0], 0])
+    i = homomorphism(F, G, [[1, 0, 0], [0, 1, 0]])
+
+    assert f == f
+    assert f != g
+    assert f != i
+    assert (f != F.identity_hom()) is False
+    assert 2*f == f*2 == homomorphism(F, F, [[2, 0], [0, 2]])
+    assert f/2 == homomorphism(F, F, [[S.Half, 0], [0, S.Half]])
+    assert f + g == homomorphism(F, F, [[1, 0], [1, x + 1]])
+    assert f - g == homomorphism(F, F, [[1, 0], [-1, 1 - x]])
+    assert f*g == g == g*f
+    assert h*g == homomorphism(F, F, [0, [1, 0]])
+    assert g*h == homomorphism(F, F, [0, 0])
+    assert i*f == i
+    assert f([1, 2]) == [1, 2]
+    assert g([1, 2]) == [2, 2*x]
+
+    assert i.restrict_domain(F.submodule([x, x]))([x, x]) == i([x, x])
+    h1 = h.quotient_domain(F.submodule([0, 1]))
+    assert h1([1, 0]) == h([1, 0])
+    assert h1.restrict_domain(h1.domain.submodule([x, 0]))([x, 0]) == h([x, 0])
+
+    raises(TypeError, lambda: f/g)
+    raises(TypeError, lambda: f + 1)
+    raises(TypeError, lambda: f + i)
+    raises(TypeError, lambda: f - 1)
+    raises(TypeError, lambda: f*i)
+
+
+def test_creation():
+    F = QQ.old_poly_ring(x).free_module(3)
+    G = QQ.old_poly_ring(x).free_module(2)
+    SM = F.submodule([1, 1, 1])
+    Q = F / SM
+    SQ = Q.submodule([1, 0, 0])
+
+    matrix = [[1, 0], [0, 1], [-1, -1]]
+    h = homomorphism(F, G, matrix)
+    h2 = homomorphism(Q, G, matrix)
+    assert h.quotient_domain(SM) == h2
+    raises(ValueError, lambda: h.quotient_domain(F.submodule([1, 0, 0])))
+    assert h2.restrict_domain(SQ) == homomorphism(SQ, G, matrix)
+    raises(ValueError, lambda: h.restrict_domain(G))
+    raises(ValueError, lambda: h.restrict_codomain(G.submodule([1, 0])))
+    raises(ValueError, lambda: h.quotient_codomain(F))
+
+    im = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
+    for M in [F, SM, Q, SQ]:
+        assert M.identity_hom() == homomorphism(M, M, im)
+    assert SM.inclusion_hom() == homomorphism(SM, F, im)
+    assert SQ.inclusion_hom() == homomorphism(SQ, Q, im)
+    assert Q.quotient_hom() == homomorphism(F, Q, im)
+    assert SQ.quotient_hom() == homomorphism(SQ.base, SQ, im)
+
+    class conv:
+        def convert(x, y=None):
+            return x
+
+    class dummy:
+        container = conv()
+
+        def submodule(*args):
+            return None
+    raises(TypeError, lambda: homomorphism(dummy(), G, matrix))
+    raises(TypeError, lambda: homomorphism(F, dummy(), matrix))
+    raises(
+        ValueError, lambda: homomorphism(QQ.old_poly_ring(x, y).free_module(3), G, matrix))
+    raises(ValueError, lambda: homomorphism(F, G, [0, 0]))
+
+
+def test_properties():
+    R = QQ.old_poly_ring(x, y)
+    F = R.free_module(2)
+    h = homomorphism(F, F, [[x, 0], [y, 0]])
+    assert h.kernel() == F.submodule([-y, x])
+    assert h.image() == F.submodule([x, 0], [y, 0])
+    assert not h.is_injective()
+    assert not h.is_surjective()
+    assert h.restrict_codomain(h.image()).is_surjective()
+    assert h.restrict_domain(F.submodule([1, 0])).is_injective()
+    assert h.quotient_domain(
+        h.kernel()).restrict_codomain(h.image()).is_isomorphism()
+
+    R2 = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
+    F = R2.free_module(2)
+    h = homomorphism(F, F, [[x, 0], [y, y + 1]])
+    assert h.is_isomorphism()

rl/Lib/site-packages/sympy/polys/agca/tests/test_ideals.py

@@ -0,0 +1,131 @@
+"""Test ideals.py code."""
+
+from sympy.polys import QQ, ilex
+from sympy.abc import x, y, z
+from sympy.testing.pytest import raises
+
+
+def test_ideal_operations():
+    R = QQ.old_poly_ring(x, y)
+    I = R.ideal(x)
+    J = R.ideal(y)
+    S = R.ideal(x*y)
+    T = R.ideal(x, y)
+
+    assert not (I == J)
+    assert I == I
+
+    assert I.union(J) == T
+    assert I + J == T
+    assert I + T == T
+
+    assert not I.subset(T)
+    assert T.subset(I)
+
+    assert I.product(J) == S
+    assert I*J == S
+    assert x*J == S
+    assert I*y == S
+    assert R.convert(x)*J == S
+    assert I*R.convert(y) == S
+
+    assert not I.is_zero()
+    assert not J.is_whole_ring()
+
+    assert R.ideal(x**2 + 1, x).is_whole_ring()
+    assert R.ideal() == R.ideal(0)
+    assert R.ideal().is_zero()
+
+    assert T.contains(x*y)
+    assert T.subset([x, y])
+
+    assert T.in_terms_of_generators(x) == [R(1), R(0)]
+
+    assert T**0 == R.ideal(1)
+    assert T**1 == T
+    assert T**2 == R.ideal(x**2, y**2, x*y)
+    assert I**5 == R.ideal(x**5)
+
+
+def test_exceptions():
+    I = QQ.old_poly_ring(x).ideal(x)
+    J = QQ.old_poly_ring(y).ideal(1)
+    raises(ValueError, lambda: I.union(x))
+    raises(ValueError, lambda: I + J)
+    raises(ValueError, lambda: I * J)
+    raises(ValueError, lambda: I.union(J))
+    assert (I == J) is False
+    assert I != J
+
+
+def test_nontriv_global():
+    R = QQ.old_poly_ring(x, y, z)
+
+    def contains(I, f):
+        return R.ideal(*I).contains(f)
+
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z)
+    assert contains([x, 1 + x + y, 5 - 7*y], 1)
+    assert contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**3)
+    assert not contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**2 + y**2)
+
+    # compare local order
+    assert not contains([x*(1 + x + y), y*(1 + z)], x)
+    assert not contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_nontriv_local():
+    R = QQ.old_poly_ring(x, y, z, order=ilex)
+
+    def contains(I, f):
+        return R.ideal(*I).contains(f)
+
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x*(1 + x + y), y*(1 + z)], x)
+    assert contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_intersection():
+    R = QQ.old_poly_ring(x, y, z)
+    # SCA, example 1.8.11
+    assert R.ideal(x, y).intersect(R.ideal(y**2, z)) == R.ideal(y**2, y*z, x*z)
+
+    assert R.ideal(x, y).intersect(R.ideal()).is_zero()
+
+    R = QQ.old_poly_ring(x, y, z, order="ilex")
+    assert R.ideal(x, y).intersect(R.ideal(y**2 + y**2*z, z + z*x**3*y)) == \
+        R.ideal(y**2, y*z, x*z)
+
+
+def test_quotient():
+    # SCA, example 1.8.13
+    R = QQ.old_poly_ring(x, y, z)
+    assert R.ideal(x, y).quotient(R.ideal(y**2, z)) == R.ideal(x, y)
+
+
+def test_reduction():
+    from sympy.polys.distributedmodules import sdm_nf_buchberger_reduced
+    R = QQ.old_poly_ring(x, y)
+    I = R.ideal(x**5, y)
+    e = R.convert(x**3 + y**2)
+    assert I.reduce_element(e) == e
+    assert I.reduce_element(e, NF=sdm_nf_buchberger_reduced) == R.convert(x**3)

rl/Lib/site-packages/sympy/polys/agca/tests/test_modules.py

@@ -0,0 +1,408 @@
+"""Test modules.py code."""
+
+from sympy.polys.agca.modules import FreeModule, ModuleOrder, FreeModulePolyRing
+from sympy.polys import CoercionFailed, QQ, lex, grlex, ilex, ZZ
+from sympy.abc import x, y, z
+from sympy.testing.pytest import raises
+from sympy.core.numbers import Rational
+
+
+def test_FreeModuleElement():
+    M = QQ.old_poly_ring(x).free_module(3)
+    e = M.convert([1, x, x**2])
+    f = [QQ.old_poly_ring(x).convert(1), QQ.old_poly_ring(x).convert(x), QQ.old_poly_ring(x).convert(x**2)]
+    assert list(e) == f
+    assert f[0] == e[0]
+    assert f[1] == e[1]
+    assert f[2] == e[2]
+    raises(IndexError, lambda: e[3])
+
+    g = M.convert([x, 0, 0])
+    assert e + g == M.convert([x + 1, x, x**2])
+    assert f + g == M.convert([x + 1, x, x**2])
+    assert -e == M.convert([-1, -x, -x**2])
+    assert e - g == M.convert([1 - x, x, x**2])
+    assert e != g
+
+    assert M.convert([x, x, x]) / QQ.old_poly_ring(x).convert(x) == [1, 1, 1]
+    R = QQ.old_poly_ring(x, order="ilex")
+    assert R.free_module(1).convert([x]) / R.convert(x) == [1]
+
+
+def test_FreeModule():
+    M1 = FreeModule(QQ.old_poly_ring(x), 2)
+    assert M1 == FreeModule(QQ.old_poly_ring(x), 2)
+    assert M1 != FreeModule(QQ.old_poly_ring(y), 2)
+    assert M1 != FreeModule(QQ.old_poly_ring(x), 3)
+    M2 = FreeModule(QQ.old_poly_ring(x, order="ilex"), 2)
+
+    assert [x, 1] in M1
+    assert [x] not in M1
+    assert [2, y] not in M1
+    assert [1/(x + 1), 2] not in M1
+
+    e = M1.convert([x, x**2 + 1])
+    X = QQ.old_poly_ring(x).convert(x)
+    assert e == [X, X**2 + 1]
+    assert e == [x, x**2 + 1]
+    assert 2*e == [2*x, 2*x**2 + 2]
+    assert e*2 == [2*x, 2*x**2 + 2]
+    assert e/2 == [x/2, (x**2 + 1)/2]
+    assert x*e == [x**2, x**3 + x]
+    assert e*x == [x**2, x**3 + x]
+    assert X*e == [x**2, x**3 + x]
+    assert e*X == [x**2, x**3 + x]
+
+    assert [x, 1] in M2
+    assert [x] not in M2
+    assert [2, y] not in M2
+    assert [1/(x + 1), 2] in M2
+
+    e = M2.convert([x, x**2 + 1])
+    X = QQ.old_poly_ring(x, order="ilex").convert(x)
+    assert e == [X, X**2 + 1]
+    assert e == [x, x**2 + 1]
+    assert 2*e == [2*x, 2*x**2 + 2]
+    assert e*2 == [2*x, 2*x**2 + 2]
+    assert e/2 == [x/2, (x**2 + 1)/2]
+    assert x*e == [x**2, x**3 + x]
+    assert e*x == [x**2, x**3 + x]
+    assert e/(1 + x) == [x/(1 + x), (x**2 + 1)/(1 + x)]
+    assert X*e == [x**2, x**3 + x]
+    assert e*X == [x**2, x**3 + x]
+
+    M3 = FreeModule(QQ.old_poly_ring(x, y), 2)
+    assert M3.convert(e) == M3.convert([x, x**2 + 1])
+
+    assert not M3.is_submodule(0)
+    assert not M3.is_zero()
+
+    raises(NotImplementedError, lambda: ZZ.old_poly_ring(x).free_module(2))
+    raises(NotImplementedError, lambda: FreeModulePolyRing(ZZ, 2))
+    raises(CoercionFailed, lambda: M1.convert(QQ.old_poly_ring(x).free_module(3)
+           .convert([1, 2, 3])))
+    raises(CoercionFailed, lambda: M3.convert(1))
+
+
+def test_ModuleOrder():
+    o1 = ModuleOrder(lex, grlex, False)
+    o2 = ModuleOrder(ilex, lex, False)
+
+    assert o1 == ModuleOrder(lex, grlex, False)
+    assert (o1 != ModuleOrder(lex, grlex, False)) is False
+    assert o1 != o2
+
+    assert o1((1, 2, 3)) == (1, (5, (2, 3)))
+    assert o2((1, 2, 3)) == (-1, (2, 3))
+
+
+def test_SubModulePolyRing_global():
+    R = QQ.old_poly_ring(x, y)
+    F = R.free_module(3)
+    Fd = F.submodule([1, 0, 0], [1, 2, 0], [1, 2, 3])
+    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
+
+    assert F == Fd
+    assert Fd == F
+    assert F != M
+    assert M != F
+    assert Fd != M
+    assert M != Fd
+    assert Fd == F.submodule(*F.basis())
+
+    assert Fd.is_full_module()
+    assert not M.is_full_module()
+    assert not Fd.is_zero()
+    assert not M.is_zero()
+    assert Fd.submodule().is_zero()
+
+    assert M.contains([x**2 + y**2 + x, 1 + y, 1])
+    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
+    assert M.contains([y**2, 1 - x*y, -x])
+
+    assert not F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
+    assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
+    assert not M.is_submodule(0)
+
+    m = F.convert([x**2 + y**2, 1, 0])
+    n = M.convert(m)
+    assert m.module is F
+    assert n.module is M
+
+    raises(ValueError, lambda: M.submodule([1, 0, 0]))
+    raises(TypeError, lambda: M.union(1))
+    raises(ValueError, lambda: M.union(R.free_module(1).submodule([x])))
+
+    assert F.submodule([x, x, x]) != F.submodule([x, x, x], order="ilex")
+
+
+def test_SubModulePolyRing_local():
+    R = QQ.old_poly_ring(x, y, order=ilex)
+    F = R.free_module(3)
+    Fd = F.submodule([1 + x, 0, 0], [1 + y, 2 + 2*y, 0], [1, 2, 3])
+    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
+
+    assert F == Fd
+    assert Fd == F
+    assert F != M
+    assert M != F
+    assert Fd != M
+    assert M != Fd
+    assert Fd == F.submodule(*F.basis())
+
+    assert Fd.is_full_module()
+    assert not M.is_full_module()
+    assert not Fd.is_zero()
+    assert not M.is_zero()
+    assert Fd.submodule().is_zero()
+
+    assert M.contains([x**2 + y**2 + x, 1 + y, 1])
+    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
+    assert M.contains([y**2, 1 - x*y, -x])
+
+    assert F.submodule([1 + x, 0, 0]) == F.submodule([1, 0, 0])
+    assert F.submodule(
+        [1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1 + x*y])) == F
+
+    raises(ValueError, lambda: M.submodule([1, 0, 0]))
+
+
+def test_SubModulePolyRing_nontriv_global():
+    R = QQ.old_poly_ring(x, y, z)
+    F = R.free_module(1)
+
+    def contains(I, f):
+        return F.submodule(*[[g] for g in I]).contains([f])
+
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**3)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y**2)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x**4 + y**3 + 2*z*y*x)
+    assert contains([x + y + z, x*y + x*z + y*z, x*y*z], x*y*z)
+    assert contains([x, 1 + x + y, 5 - 7*y], 1)
+    assert contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**3)
+    assert not contains(
+        [x**3 + y**3, y**3 + z**3, z**3 + x**3, x**2*y + x**2*z + y**2*z],
+        x**2 + y**2)
+
+    # compare local order
+    assert not contains([x*(1 + x + y), y*(1 + z)], x)
+    assert not contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_SubModulePolyRing_nontriv_local():
+    R = QQ.old_poly_ring(x, y, z, order=ilex)
+    F = R.free_module(1)
+
+    def contains(I, f):
+        return F.submodule(*[[g] for g in I]).contains([f])
+
+    assert contains([x, y], x)
+    assert contains([x, y], x + y)
+    assert not contains([x, y], 1)
+    assert not contains([x, y], z)
+    assert contains([x**2 + y, x**2 + x], x - y)
+    assert not contains([x + y + z, x*y + x*z + y*z, x*y*z], x**2)
+    assert contains([x*(1 + x + y), y*(1 + z)], x)
+    assert contains([x*(1 + x + y), y*(1 + z)], x + y)
+
+
+def test_syzygy():
+    R = QQ.old_poly_ring(x, y, z)
+    M = R.free_module(1).submodule([x*y], [y*z], [x*z])
+    S = R.free_module(3).submodule([0, x, -y], [z, -x, 0])
+    assert M.syzygy_module() == S
+
+    M2 = M / ([x*y*z],)
+    S2 = R.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
+    assert M2.syzygy_module() == S2
+
+    F = R.free_module(3)
+    assert F.submodule(*F.basis()).syzygy_module() == F.submodule()
+
+    R2 = QQ.old_poly_ring(x, y, z) / [x*y*z]
+    M3 = R2.free_module(1).submodule([x*y], [y*z], [x*z])
+    S3 = R2.free_module(3).submodule([z, 0, 0], [0, x, 0], [0, 0, y])
+    assert M3.syzygy_module() == S3
+
+
+def test_in_terms_of_generators():
+    R = QQ.old_poly_ring(x, order="ilex")
+    M = R.free_module(2).submodule([2*x, 0], [1, 2])
+    assert M.in_terms_of_generators(
+        [x, x]) == [R.convert(Rational(1, 4)), R.convert(x/2)]
+    raises(ValueError, lambda: M.in_terms_of_generators([1, 0]))
+
+    M = R.free_module(2) / ([x, 0], [1, 1])
+    SM = M.submodule([1, x])
+    assert SM.in_terms_of_generators([2, 0]) == [R.convert(-2/(x - 1))]
+
+    R = QQ.old_poly_ring(x, y) / [x**2 - y**2]
+    M = R.free_module(2)
+    SM = M.submodule([x, 0], [0, y])
+    assert SM.in_terms_of_generators(
+        [x**2, x**2]) == [R.convert(x), R.convert(y)]
+
+
+def test_QuotientModuleElement():
+    R = QQ.old_poly_ring(x)
+    F = R.free_module(3)
+    N = F.submodule([1, x, x**2])
+    M = F/N
+    e = M.convert([x**2, 2, 0])
+
+    assert M.convert([x + 1, x**2 + x, x**3 + x**2]) == 0
+    assert e == [x**2, 2, 0] + N == F.convert([x**2, 2, 0]) + N == \
+        M.convert(F.convert([x**2, 2, 0]))
+
+    assert M.convert([x**2 + 1, 2*x + 2, x**2]) == e + [0, x, 0] == \
+        e + M.convert([0, x, 0]) == e + F.convert([0, x, 0])
+    assert M.convert([x**2 + 1, 2, x**2]) == e - [0, x, 0] == \
+        e - M.convert([0, x, 0]) == e - F.convert([0, x, 0])
+    assert M.convert([0, 2, 0]) == M.convert([x**2, 4, 0]) - e == \
+        [x**2, 4, 0] - e == F.convert([x**2, 4, 0]) - e
+    assert M.convert([x**3 + x**2, 2*x + 2, 0]) == (1 + x)*e == \
+        R.convert(1 + x)*e == e*(1 + x) == e*R.convert(1 + x)
+    assert -e == [-x**2, -2, 0]
+
+    f = [x, x, 0] + N
+    assert M.convert([1, 1, 0]) == f / x == f / R.convert(x)
+
+    M2 = F/[(2, 2*x, 2*x**2), (0, 0, 1)]
+    G = R.free_module(2)
+    M3 = G/[[1, x]]
+    M4 = F.submodule([1, x, x**2], [1, 0, 0]) / N
+    raises(CoercionFailed, lambda: M.convert(G.convert([1, x])))
+    raises(CoercionFailed, lambda: M.convert(M3.convert([1, x])))
+    raises(CoercionFailed, lambda: M.convert(M2.convert([1, x, x])))
+    assert M2.convert(M.convert([2, x, x**2])) == [2, x, 0]
+    assert M.convert(M4.convert([2, 0, 0])) == [2, 0, 0]
+
+
+def test_QuotientModule():
+    R = QQ.old_poly_ring(x)
+    F = R.free_module(3)
+    N = F.submodule([1, x, x**2])
+    M = F/N
+
+    assert M != F
+    assert M != N
+    assert M == F / [(1, x, x**2)]
+    assert not M.is_zero()
+    assert (F / F.basis()).is_zero()
+
+    SQ = F.submodule([1, x, x**2], [2, 0, 0]) / N
+    assert SQ == M.submodule([2, x, x**2])
+    assert SQ != M.submodule([2, 1, 0])
+    assert SQ != M
+    assert M.is_submodule(SQ)
+    assert not SQ.is_full_module()
+
+    raises(ValueError, lambda: N/F)
+    raises(ValueError, lambda: F.submodule([2, 0, 0]) / N)
+    raises(ValueError, lambda: R.free_module(2)/F)
+    raises(CoercionFailed, lambda: F.convert(M.convert([1, x, x**2])))
+
+    M1 = F / [[1, 1, 1]]
+    M2 = M1.submodule([1, 0, 0], [0, 1, 0])
+    assert M1 == M2
+
+
+def test_ModulesQuotientRing():
+    R = QQ.old_poly_ring(x, y, order=(("lex", x), ("ilex", y))) / [x**2 + 1]
+    M1 = R.free_module(2)
+    assert M1 == R.free_module(2)
+    assert M1 != QQ.old_poly_ring(x).free_module(2)
+    assert M1 != R.free_module(3)
+
+    assert [x, 1] in M1
+    assert [x] not in M1
+    assert [1/(R.convert(x) + 1), 2] in M1
+    assert [1, 2/(1 + y)] in M1
+    assert [1, 2/y] not in M1
+
+    assert M1.convert([x**2, y]) == [-1, y]
+
+    F = R.free_module(3)
+    Fd = F.submodule([x**2, 0, 0], [1, 2, 0], [1, 2, 3])
+    M = F.submodule([x**2 + y**2, 1, 0], [x, y, 1])
+
+    assert F == Fd
+    assert Fd == F
+    assert F != M
+    assert M != F
+    assert Fd != M
+    assert M != Fd
+    assert Fd == F.submodule(*F.basis())
+
+    assert Fd.is_full_module()
+    assert not M.is_full_module()
+    assert not Fd.is_zero()
+    assert not M.is_zero()
+    assert Fd.submodule().is_zero()
+
+    assert M.contains([x**2 + y**2 + x, -x**2 + y, 1])
+    assert not M.contains([x**2 + y**2 + x, 1 + y, 2])
+    assert M.contains([y**2, 1 - x*y, -x])
+
+    assert F.submodule([x, 0, 0]) == F.submodule([1, 0, 0])
+    assert not F.submodule([y, 0, 0]) == F.submodule([1, 0, 0])
+    assert F.submodule([1, 0, 0], [0, 1, 0]).union(F.submodule([0, 0, 1])) == F
+    assert not M.is_submodule(0)
+
+
+def test_module_mul():
+    R = QQ.old_poly_ring(x)
+    M = R.free_module(2)
+    S1 = M.submodule([x, 0], [0, x])
+    S2 = M.submodule([x**2, 0], [0, x**2])
+    I = R.ideal(x)
+
+    assert I*M == M*I == S1 == x*M == M*x
+    assert I*S1 == S2 == x*S1
+
+
+def test_intersection():
+    # SCA, example 2.8.5
+    F = QQ.old_poly_ring(x, y).free_module(2)
+    M1 = F.submodule([x, y], [y, 1])
+    M2 = F.submodule([0, y - 1], [x, 1], [y, x])
+    I = F.submodule([x, y], [y**2 - y, y - 1], [x*y + y, x + 1])
+    I1, rel1, rel2 = M1.intersect(M2, relations=True)
+    assert I1 == M2.intersect(M1) == I
+    for i, g in enumerate(I1.gens):
+        assert g == sum(c*x for c, x in zip(rel1[i], M1.gens)) \
+                 == sum(d*y for d, y in zip(rel2[i], M2.gens))
+
+    assert F.submodule([x, y]).intersect(F.submodule([y, x])).is_zero()
+
+
+def test_quotient():
+    # SCA, example 2.8.6
+    R = QQ.old_poly_ring(x, y, z)
+    F = R.free_module(2)
+    assert F.submodule([x*y, x*z], [y*z, x*y]).module_quotient(
+        F.submodule([y, z], [z, y])) == QQ.old_poly_ring(x, y, z).ideal(x**2*y**2 - x*y*z**2)
+    assert F.submodule([x, y]).module_quotient(F.submodule()).is_whole_ring()
+
+    M = F.submodule([x**2, x**2], [y**2, y**2])
+    N = F.submodule([x + y, x + y])
+    q, rel = M.module_quotient(N, relations=True)
+    assert q == R.ideal(y**2, x - y)
+    for i, g in enumerate(q.gens):
+        assert g*N.gens[0] == sum(c*x for c, x in zip(rel[i], M.gens))
+
+
+def test_groebner_extendend():
+    M = QQ.old_poly_ring(x, y, z).free_module(3).submodule([x + 1, y, 1], [x*y, z, z**2])
+    G, R = M._groebner_vec(extended=True)
+    for i, g in enumerate(G):
+        assert g == sum(c*gen for c, gen in zip(R[i], M.gens))