def _single_variate(base_ring, names, sparse):
    """
    EXAMPLES::

        sage: from sage.rings.polynomial.laurent_polynomial_ring import _single_variate
        sage: _single_variate(QQ, ('x',), False)
        Univariate Laurent Polynomial Ring in x over Rational Field
    """
    ############################################################
    # This should later get moved to an actual single variate  #
    # implementation with valuation tracking,                  #
    # but I don't want to right now.                           #
    ############################################################
    # We need to come up with a name for the inverse that is easy to search
    # for in a string *and* doesn't overlap with the name that we already have.
    # For now, I'm going to use a name mangling with checking method.
    names = normalize_names(1, names)
    key = (base_ring, names, sparse)
    P = _get_from_cache(key)
    if P is not None:
        return P
    prepend_string = "qk"
    while True:
        if prepend_string in names:
            prepend_string += 'k'
        else:
            break
    R = _multi_variate_poly(base_ring, names, 1, sparse, 'degrevlex', None)
    P = LaurentPolynomialRing_mpair(R, prepend_string, names)
    _save_in_cache(key, P)
    return P

def AffineSpace(n, R=None, names='x'):
    r"""
    Return affine space of dimension `n` over the ring `R`.

    EXAMPLES:

    The dimension and ring can be given in either order::

        sage: AffineSpace(3, QQ, 'x')
        Affine Space of dimension 3 over Rational Field
        sage: AffineSpace(5, QQ, 'x')
        Affine Space of dimension 5 over Rational Field
        sage: A = AffineSpace(2, QQ, names='XY'); A
        Affine Space of dimension 2 over Rational Field
        sage: A.coordinate_ring()
        Multivariate Polynomial Ring in X, Y over Rational Field

    Use the divide operator for base extension::

        sage: AffineSpace(5, names='x')/GF(17)
        Affine Space of dimension 5 over Finite Field of size 17

    The default base ring is `\ZZ`::

        sage: AffineSpace(5, names='x')
        Affine Space of dimension 5 over Integer Ring

    There is also an affine space associated to each polynomial ring::

        sage: R = GF(7)['x,y,z']
        sage: A = AffineSpace(R); A
        Affine Space of dimension 3 over Finite Field of size 7
        sage: A.coordinate_ring() is R
        True
    """
    if is_MPolynomialRing(n) and R is None:
        R = n
        A = AffineSpace(R.ngens(), R.base_ring(), R.variable_names())
        A._coordinate_ring = R
        return A
    if isinstance(R, (int, long, Integer)):
        n, R = R, n
    if R is None:
        R = ZZ  # default is the integers
    if names is None:
        if n == 0:
            names = ''
        else:
            raise TypeError("You must specify the variables names of the coordinate ring.")
    names = normalize_names(n, names)
    if R in _Fields:
        if is_FiniteField(R):
            return AffineSpace_finite_field(n, R, names)
        else:
            return AffineSpace_field(n, R, names)
    return AffineSpace_generic(n, R, names)

    def __init__(self, n, R=ZZ, names=None):
        """
        EXAMPLES::

            sage: ProjectiveSpace(3, Zp(5), 'y')
            Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
        """
        names = normalize_names(n+1, names)
        AmbientSpace.__init__(self, n, R)
        self._assign_names(names)

    def __init__(self, n, R, names):
        """
        EXAMPLES::

            sage: AffineSpace(3, Zp(5), 'y')
            Affine Space of dimension 3 over 5-adic Ring with capped relative precision 20
        """
        names = normalize_names(n, names)
        AmbientSpace.__init__(self, n, R)
        self._assign_names(names)

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), 'conway', None, '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), 'conway', None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        from sage.structure.proof.all import WithProof, arithmetic

        if proof is None:
            proof = arithmetic()
        with WithProof("arithmetic", proof):
            order = int(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be > 1.")

            if arith.is_prime(order):
                name = None
                modulus = None
                p = integer.Integer(order)
                n = integer.Integer(1)
            elif arith.is_prime_power(order):
                if not names is None:
                    name = names
                name = normalize_names(1, name)

                p, n = arith.factor(order)[0]

                if modulus is None or modulus == "default":
                    if exists_conway_polynomial(p, n):
                        modulus = "conway"
                    else:
                        if p == 2:
                            modulus = "minimal_weight"
                        else:
                            modulus = "random"
                elif modulus == "random":
                    modulus += str(random.randint(0, 1 << 128))

                if isinstance(modulus, (list, tuple)):
                    modulus = FiniteField(p)["x"](modulus)
                # some classes use 'random' as the modulus to
                # generate a random modulus, but we don't want
                # to cache it
                elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
                    modulus = modulus.change_variable_name("x")
                elif not isinstance(modulus, str):
                    raise ValueError("Modulus parameter not understood.")
            else:  # Neither a prime, nor a prime power
                raise ValueError("the order of a finite field must be a prime power.")

            return (order, name, modulus, impl, str(kwds), p, n, proof), kwds

def _single_variate(base_ring, name, sparse, implementation):
    import sage.rings.polynomial.polynomial_ring as m

    name = normalize_names(1, name)
    key = (base_ring, name, sparse, implementation if not sparse else None)
    R = _get_from_cache(key)
    if not R is None:
        return R

    if isinstance(base_ring, ring.CommutativeRing):
        if is_IntegerModRing(base_ring) and not sparse:
            n = base_ring.order()
            if n.is_prime():
                R = m.PolynomialRing_dense_mod_p(base_ring, name, implementation=implementation)
            elif n > 1:
                R = m.PolynomialRing_dense_mod_n(base_ring, name, implementation=implementation)
            else:  # n == 1!
                R = m.PolynomialRing_integral_domain(base_ring, name)  # specialized code breaks in this case.

        elif is_FiniteField(base_ring) and not sparse:
            R = m.PolynomialRing_dense_finite_field(base_ring, name, implementation=implementation)

        elif isinstance(base_ring, padic_base_leaves.pAdicFieldCappedRelative):
            R = m.PolynomialRing_dense_padic_field_capped_relative(base_ring, name)

        elif isinstance(base_ring, padic_base_leaves.pAdicRingCappedRelative):
            R = m.PolynomialRing_dense_padic_ring_capped_relative(base_ring, name)

        elif isinstance(base_ring, padic_base_leaves.pAdicRingCappedAbsolute):
            R = m.PolynomialRing_dense_padic_ring_capped_absolute(base_ring, name)

        elif isinstance(base_ring, padic_base_leaves.pAdicRingFixedMod):
            R = m.PolynomialRing_dense_padic_ring_fixed_mod(base_ring, name)

        elif base_ring.is_field(proof=False):
            R = m.PolynomialRing_field(base_ring, name, sparse)

        elif base_ring.is_integral_domain(proof=False):
            R = m.PolynomialRing_integral_domain(base_ring, name, sparse, implementation)
        else:
            R = m.PolynomialRing_commutative(base_ring, name, sparse)
    else:
        R = m.PolynomialRing_general(base_ring, name, sparse)

    if hasattr(R, "_implementation_names"):
        for name in R._implementation_names:
            real_key = key[0:3] + (name,)
            _save_in_cache(real_key, R)
    else:
        _save_in_cache(key, R)
    return R

def FreeAbelianMonoid(index_set=None, names=None, **kwds):
    """
    Return a free abelian monoid on `n` generators or with the generators
    indexed by a set `I`.

    We construct free abelian monoids by specifing either:

    - the number of generators and/or the names of the generators
    - the indexing set for the generators (this ignores the other two inputs)

    INPUT:

    - ``index_set`` -- an indexing set for the generators; if an integer,
      then this becomes `\{0, 1, \ldots, n-1\}`

    -  ``names`` -- names of generators

    OUTPUT:

    A free abelian monoid.

    EXAMPLES::

        sage: F.<a,b,c,d,e> = FreeAbelianMonoid(); F
        Free abelian monoid on 5 generators (a, b, c, d, e)
        sage: FreeAbelianMonoid(index_set=ZZ)
        Free abelian monoid indexed by Integer Ring
    """
    if isinstance(index_set, str): # Swap args (this works if names is None as well)
        names, index_set = index_set, names

    if index_set is None and names is not None:
        if isinstance(names, str):
            index_set = names.count(',')
        else:
            index_set = len(names)

    if index_set not in ZZ:
        if names is not None:
            names = normalize_names(len(names), names)
        from sage.monoids.indexed_free_monoid import IndexedFreeAbelianMonoid
        return IndexedFreeAbelianMonoid(index_set, names=names, **kwds)

    if names is None:
        raise ValueError("names must be specified")
    return FreeAbelianMonoid_factory(index_set, names)

def _multi_variate(base_ring, names, n, sparse, order, implementation):
    #    if not sparse:
    #        raise ValueError, "A dense representation of multivariate polynomials is not supported"
    sparse = False
    # "True" would be correct, since there is no dense implementation of
    # multivariate polynomials. However, traditionally, "False" is used in the key,
    # even though it is meaningless.

    if implementation is not None:
        raise ValueError("The %s implementation is not known for multivariate polynomial rings" % implementation)

    names = normalize_names(n, names)

    import sage.rings.polynomial.multi_polynomial_ring as m
    from sage.rings.polynomial.term_order import TermOrder

    order = TermOrder(order, n)

    key = (base_ring, names, n, sparse, order)
    R = _get_from_cache(key)
    if not R is None:
        return R

    from sage.rings.polynomial.multi_polynomial_libsingular import MPolynomialRing_libsingular

    if m.integral_domain.is_IntegralDomain(base_ring):
        if n < 1:
            R = m.MPolynomialRing_polydict_domain(base_ring, n, names, order)
        else:
            try:
                R = MPolynomialRing_libsingular(base_ring, n, names, order)
            except (TypeError, NotImplementedError):
                R = m.MPolynomialRing_polydict_domain(base_ring, n, names, order)
    else:
        if not base_ring.is_zero():
            try:
                R = MPolynomialRing_libsingular(base_ring, n, names, order)
            except (TypeError, NotImplementedError):
                R = m.MPolynomialRing_polydict(base_ring, n, names, order)
        else:
            R = m.MPolynomialRing_polydict(base_ring, n, names, order)
    _save_in_cache(key, R)
    return R

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None,
                                  impl=None, proof=None, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        from sage.structure.proof.all import WithProof, arithmetic
        if proof is None: proof = arithmetic()
        with WithProof('arithmetic', proof):
            order = int(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be > 1.")

            if arith.is_prime(order):
                name = None
                modulus = None
                p = integer.Integer(order)
                n = integer.Integer(1)
            elif arith.is_prime_power(order):
                if not names is None: name = names
                name = normalize_names(1,name)

                p, n = arith.factor(order)[0]

                if modulus is None or isinstance(modulus, str):
                    # A string specifies an algorithm to find a suitable modulus.
                    if modulus == "default":    # for backward compatibility
                        modulus = None
                    modulus = GF(p)['x'].irreducible_element(n, algorithm=modulus)
                elif isinstance(modulus, (list, tuple)):
                    modulus = GF(p)['x'](modulus)
                elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
                    modulus = modulus.change_variable_name('x')
                else:
                    raise TypeError("wrong type for modulus parameter")
            else:
                raise ValueError("the order of a finite field must be a prime power.")

            return (order, name, modulus, impl, str(kwds), p, n, proof), kwds

def _multi_variate(base_ring, names, n, sparse, order):
    """
    EXAMPLES::

        sage: from sage.rings.polynomial.laurent_polynomial_ring import _multi_variate
        sage: _multi_variate(QQ, ('x','y'), 2, False, 'degrevlex')
        Multivariate Laurent Polynomial Ring in x, y over Rational Field
    """
    # We need to come up with a name for the inverse that is easy to search
    # for in a string *and* doesn't overlap with the name that we already have.
    # For now, I'm going to use a name mangling with checking method.
    names = normalize_names(n, names)

    from term_order import TermOrder
    order = TermOrder(order, n)

    if isinstance(names, list):
        names = tuple(names)
    elif isinstance(names, str):
        if ',' in names:
            names = tuple(names.split(','))

    key = (base_ring, names, n, sparse, order)
    P = _get_from_cache(key)
    if P is not None:
        return P
    prepend_string = "qk"
    while True:
        for a in names:
            if prepend_string in a:
                prepend_string += 'k'
                break
        else:
            break
    R = _multi_variate_poly(base_ring, names, n, sparse, order, None)
    P = LaurentPolynomialRing_mpair(R, prepend_string, names)
    _save_in_cache(key, P)
    return P

def _single_variate(base_ring, names, sparse):
    """
    EXAMPLES::

        sage: from sage.rings.polynomial.laurent_polynomial_ring import _single_variate
        sage: _single_variate(QQ, ('x',), False)
        Univariate Laurent Polynomial Ring in x over Rational Field
    """
    names = normalize_names(1, names)
    key = (base_ring, names, sparse)
    P = _get_from_cache(key)
    if P is not None:
        return P
    prepend_string = "qk"
    while True:
        if prepend_string in names:
            prepend_string += 'k'
        else:
            break
    R = _single_variate_poly(base_ring, names, sparse, None)
    P = LaurentPolynomialRing_univariate(R, names)
    _save_in_cache(key, P)
    return P

 def create_key(self, n, names):
     n = int(n)
     names = normalize_names(n, names)
     return (n, names)

def BooleanPolynomialRing_constructor(n=None, names=None, order="lex"):
    """
    Construct a boolean polynomial ring with the following
    parameters:

    INPUT:

    - ``n`` -- number of variables (an integer > 1)
    - ``names`` -- names of ring variables, may be a string or list/tuple of strings
    - ``order`` -- term order (default: lex)

    EXAMPLES::

        sage: R.<x, y, z> = BooleanPolynomialRing() # indirect doctest
        sage: R
        Boolean PolynomialRing in x, y, z

        sage: p = x*y + x*z + y*z
        sage: x*p
        x*y*z + x*y + x*z

        sage: R.term_order()
        Lexicographic term order

        sage: R = BooleanPolynomialRing(5,'x',order='deglex(3),deglex(2)')
        sage: R.term_order()
        Block term order with blocks:
        (Degree lexicographic term order of length 3,
         Degree lexicographic term order of length 2)

        sage: R = BooleanPolynomialRing(3,'x',order='degrevlex')
        sage: R.term_order()
        Degree reverse lexicographic term order

        sage: BooleanPolynomialRing(names=('x','y'))
        Boolean PolynomialRing in x, y

        sage: BooleanPolynomialRing(names='x,y')
        Boolean PolynomialRing in x, y

    TESTS::

        sage: P.<x,y> = BooleanPolynomialRing(2,order='degrevlex')
        sage: x > y
        True

        sage: P.<x0, x1, x2, x3> = BooleanPolynomialRing(4,order='degrevlex(2),degrevlex(2)')
        sage: x0 > x1
        True
        sage: x2 > x3
        True
    """

    if isinstance(n, str):
        names = n
        n = 0
    if n is None and names is not None:
        n = 0

    names = normalize_names(n, names)
    if n is 0:
        n = len(names)

    from sage.rings.polynomial.term_order import TermOrder

    order = TermOrder(order, n)

    key = ("pbori", names, n, order)
    R = _get_from_cache(key)
    if not R is None:
        return R

    from sage.rings.polynomial.pbori import BooleanPolynomialRing

    R = BooleanPolynomialRing(n, names, order)

    _save_in_cache(key, R)
    return R

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None,
                                  impl=None, proof=None, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        from sage.structure.proof.all import WithProof, arithmetic
        if proof is None: proof = arithmetic()
        with WithProof('arithmetic', proof):
            order = int(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be > 1.")

            if arith.is_prime(order):
                name = None
                modulus = None
                p = integer.Integer(order)
                n = integer.Integer(1)
            elif arith.is_prime_power(order):
                if not names is None: name = names
                name = normalize_names(1,name)

                p, n = arith.factor(order)[0]

                # The following is a temporary solution that allows us
                # to construct compatible systems of finite fields
                # until algebraic closures of finite fields are
                # implemented in Sage.  It requires the user to
                # specify two parameters:
                #
                # - `conway` -- boolean; if True, this field is
                #   constructed to fit in a compatible system using
                #   a Conway polynomial.
                # - `prefix` -- a string used to generate names for
                #   automatically constructed finite fields
                #
                # See the docstring of FiniteFieldFactory for examples.
                #
                # Once algebraic closures of finite fields are
                # implemented, this syntax should be superseded by
                # something like the following:
                #
                #     sage: Fpbar = GF(5).algebraic_closure('z')
                #     sage: F, e = Fpbar.subfield(3)  # e is the embedding into Fpbar
                #     sage: F
                #     Finite field in z3 of size 5^3
                #
                # This temporary solution only uses actual Conway
                # polynomials (no pseudo-Conway polynomials), since
                # pseudo-Conway polynomials are not unique, and until
                # we have algebraic closures of finite fields, there
                # is no good place to store a specific choice of
                # pseudo-Conway polynomials.
                if name is None:
                    if not (kwds.has_key('conway') and kwds['conway']):
                        raise ValueError("parameter 'conway' is required if no name given")
                    if not kwds.has_key('prefix'):
                        raise ValueError("parameter 'prefix' is required if no name given")
                    name = kwds['prefix'] + str(n)

                if kwds.has_key('conway') and kwds['conway']:
                    from conway_polynomials import conway_polynomial
                    if not kwds.has_key('prefix'):
                        raise ValueError("a prefix must be specified if conway=True")
                    if modulus is not None:
                        raise ValueError("no modulus may be specified if conway=True")
                    # The following raises a RuntimeError if no polynomial is found.
                    modulus = conway_polynomial(p, n)

                if modulus is None or isinstance(modulus, str):
                    # A string specifies an algorithm to find a suitable modulus.
                    if modulus == "default":    # for backward compatibility
                        modulus = None
                    modulus = GF(p)['x'].irreducible_element(n, algorithm=modulus)
                elif isinstance(modulus, (list, tuple)):
                    modulus = GF(p)['x'](modulus)
                elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
                    modulus = modulus.change_variable_name('x')
                else:
                    raise TypeError("wrong type for modulus parameter")
            else:
                raise ValueError("the order of a finite field must be a prime power.")

            return (order, name, modulus, impl, str(kwds), p, n, proof), kwds

#.........这里部分代码省略.........
        sage: U.term_order()
        Negative degree lexicographic term order
        sage: U(p)
        -b^2 - a*b^3 - 2*b^6 + a^7 + a^5*b^2 + O(a, b)^9


    TESTS::

        sage: N = PowerSeriesRing(QQ,'k',num_gens=5); N
        Multivariate Power Series Ring in k0, k1, k2, k3, k4 over Rational Field

    The following behavior of univariate power series ring will eventually
    be deprecated and then changed to return a multivariate power series
    ring::

        sage: N = PowerSeriesRing(QQ,'k',5); N
        Power Series Ring in k over Rational Field
        sage: N.default_prec()
        5
        sage: L.<m> = PowerSeriesRing(QQ,5); L
        Power Series Ring in m over Rational Field
        sage: L.default_prec()
        5

    """
    #multivariate case:
    # examples for first case:
    # PowerSeriesRing(QQ,'x,y,z')
    # PowerSeriesRing(QQ,['x','y','z'])
    # PowerSeriesRing(QQ,['x','y','z'], 3)
    if names is None and name is not None:
        names = name
    if isinstance(names, (tuple, list)) and len(names) > 1 or (isinstance(names, str) and ',' in names):
        return _multi_variate(base_ring, num_gens=arg2, names=names,
                     order=order, default_prec=default_prec, sparse=sparse)
    # examples for second case:
    # PowerSeriesRing(QQ,3,'t')
    if arg2 is None and num_gens is not None:
        arg2 = names
        names = num_gens
    if isinstance(arg2, str) and isinstance(names, (int,long,integer.Integer)):
        return _multi_variate(base_ring, num_gens=names, names=arg2,
                     order=order, default_prec=default_prec, sparse=sparse)

    
    # univariate case: the arguments to PowerSeriesRing used to be
    # (base_ring, name=None, default_prec=20, names=None, sparse=False),
    # and thus that is what the code below expects; this behavior is being
    # deprecated, and will eventually be removed.
    if default_prec is None and arg2 is None:
        default_prec = 20
    elif arg2 is not None:
        default_prec = arg2

    ## too many things (padics, elliptic curves) depend on this behavior,
    ## so no warning for now.
    ##
    # from sage.misc.misc import deprecation
    # if isinstance(name, (int,long,integer.Integer)) or isinstance(arg2,(int,long,integer.Integer)):
    #     deprecation("This behavior of PowerSeriesRing is being deprecated in favor of constructing multivariate power series rings. (See Trac ticket #1956.)")


    # the following is the original, univariate-only code

    if isinstance(name, (int,long,integer.Integer)):
        default_prec = name
    if not names is None:
        name = names
    try:
        name = gens.normalize_names(1, name)
    except TypeError:
        raise TypeError, "illegal variable name"
    
    if name is None:
        raise TypeError, "You must specify the name of the indeterminate of the Power series ring."
    
    key = (base_ring, name, default_prec, sparse)
    if _cache.has_key(key):
        R = _cache[key]()
        if not R is None:
            return R

    if isinstance(name, (tuple, list)):
        assert len(name) == 1
        name = name[0]

    if not (name is None or isinstance(name, str)):
        raise TypeError, "variable name must be a string or None"
        
                  
    if isinstance(base_ring, field.Field):
        R = PowerSeriesRing_over_field(base_ring, name, default_prec, sparse=sparse)
    elif isinstance(base_ring, integral_domain.IntegralDomain):
        R = PowerSeriesRing_domain(base_ring, name, default_prec, sparse=sparse)
    elif isinstance(base_ring, commutative_ring.CommutativeRing):
        R = PowerSeriesRing_generic(base_ring, name, default_prec, sparse=sparse)
    else:
        raise TypeError, "base_ring must be a commutative ring"
    _cache[key] = weakref.ref(R)
    return R

def FreeMonoid(index_set=None, names=None, commutative=False, **kwds):
    r"""
    Return a free monoid on `n` generators or with the generators indexed by
    a set `I`.

    We construct free monoids by specifing either:

    - the number of generators and/or the names of the generators
    - the indexing set for the generators

    INPUT:

    - ``index_set`` -- an indexing set for the generators; if an integer,
      than this becomes `\{0, 1, \ldots, n-1\}`

    -  ``names`` -- names of generators

    - ``commutative`` -- (default: ``False``) whether the free monoid is
      commutative or not

    OUTPUT:

    A free monoid.

    EXAMPLES::

        sage: F.<a,b,c,d,e> = FreeMonoid(); F
        Free monoid on 5 generators (a, b, c, d, e)
        sage: FreeMonoid(index_set=ZZ)
        Free monoid indexed by Integer Ring

        sage: F.<x,y,z> = FreeMonoid(abelian=True); F
        Free abelian monoid on 3 generators (x, y, z)
        sage: FreeMonoid(index_set=ZZ, commutative=True)
        Free abelian monoid indexed by Integer Ring
    """
    if 'abelian' in kwds:
        commutative = kwds['abelian']
        del kwds['abelian']

    if commutative:
        from sage.monoids.free_abelian_monoid import FreeAbelianMonoid
        return FreeAbelianMonoid(index_set, names, **kwds)

    if isinstance(index_set, str): # Swap args (this works if names is None as well)
        names, index_set = index_set, names

    if index_set is None and names is not None:
        if isinstance(names, str):
            index_set = names.count(',')
        else:
            index_set = len(names)

    if index_set not in ZZ:
        if names is not None:
            names = normalize_names(len(names), names)
        from sage.monoids.indexed_free_monoid import IndexedFreeMonoid
        return IndexedFreeMonoid(index_set, names=names, **kwds)

    if names is None:
        raise ValueError("names must be specified")
    return FreeMonoid_factory(index_set, names)

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None,
                                  impl=None, proof=None, check_irreducible=True, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), x^2 + 2*x + 2, 'givaro', "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        import sage.rings.arith
        from sage.structure.proof.all import WithProof, arithmetic
        if proof is None:
            proof = arithmetic()
        with WithProof('arithmetic', proof):
            order = Integer(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be at least 2")

            if order.is_prime():
                p = order
                n = Integer(1)
                if impl is None:
                    impl = 'modn'
                name = ('x',)  # Ignore name
                # Every polynomial of degree 1 is irreducible
                check_irreducible = False
            # This check should be replaced by order.is_prime_power()
            # if Trac #16878 is fixed.
            elif sage.rings.arith.is_prime_power(order):
                if not names is None: name = names
                name = normalize_names(1,name)

                p, n = order.factor()[0]

                # The following is a temporary solution that allows us
                # to construct compatible systems of finite fields
                # until algebraic closures of finite fields are
                # implemented in Sage.  It requires the user to
                # specify two parameters:
                #
                # - `conway` -- boolean; if True, this field is
                #   constructed to fit in a compatible system using
                #   a Conway polynomial.
                # - `prefix` -- a string used to generate names for
                #   automatically constructed finite fields
                #
                # See the docstring of FiniteFieldFactory for examples.
                #
                # Once algebraic closures of finite fields are
                # implemented, this syntax should be superseded by
                # something like the following:
                #
                #     sage: Fpbar = GF(5).algebraic_closure('z')
                #     sage: F, e = Fpbar.subfield(3)  # e is the embedding into Fpbar
                #     sage: F
                #     Finite field in z3 of size 5^3
                #
                # This temporary solution only uses actual Conway
                # polynomials (no pseudo-Conway polynomials), since
                # pseudo-Conway polynomials are not unique, and until
                # we have algebraic closures of finite fields, there
                # is no good place to store a specific choice of
                # pseudo-Conway polynomials.
                if name is None:
                    if not ('conway' in kwds and kwds['conway']):
                        raise ValueError("parameter 'conway' is required if no name given")
                    if 'prefix' not in kwds:
                        raise ValueError("parameter 'prefix' is required if no name given")
                    name = kwds['prefix'] + str(n)

                if 'conway' in kwds and kwds['conway']:
                    from conway_polynomials import conway_polynomial
                    if 'prefix' not in kwds:
                        raise ValueError("a prefix must be specified if conway=True")
                    if modulus is not None:
                        raise ValueError("no modulus may be specified if conway=True")
                    # The following raises a RuntimeError if no polynomial is found.
                    modulus = conway_polynomial(p, n)

                if impl is None:
                    if order < zech_log_bound:
                        impl = 'givaro'
                    elif p == 2:
                        impl = 'ntl'
                    else:
                        impl = 'pari_ffelt'
            else:
                raise ValueError("the order of a finite field must be a prime power")

            # Determine modulus.
            # For the 'modn' implementation, we use the following
            # optimization which we also need to avoid an infinite loop:
            # a modulus of None is a shorthand for x-1.
            if modulus is not None or impl != 'modn':
                R = PolynomialRing(FiniteField(p), 'x')
                if modulus is None:
                    modulus = R.irreducible_element(n)
                if isinstance(modulus, str):
                    # A string specifies an algorithm to find a suitable modulus.
#.........这里部分代码省略.........

def ProductProjectiveSpaces(n, R=None, names='x'):
    r"""
    Returns the cartesian product of projective spaces. Can input either a list of projective spaces
    over the same base ring or the list of dimensions, the base ring, and the variable names.

    INPUT:

    - ``n`` -- a list of integers or a list of projective spaces

    - ``R`` -- a ring

    - ``names`` -- a string or list of strings

    EXAMPLES::

        sage: P1 = ProjectiveSpace(QQ,2,'x')
        sage: P2 = ProjectiveSpace(QQ,3,'y')
        sage: ProductProjectiveSpaces([P1,P2])
        Product of projective spaces P^2 x P^3 over Rational Field

    ::

        sage: ProductProjectiveSpaces([2,2],GF(7),'y')
        Product of projective spaces P^2 x P^2 over Finite Field of size 7

    ::

        sage: P1 = ProjectiveSpace(ZZ,2,'x')
        sage: P2 = ProjectiveSpace(QQ,3,'y')
        sage: ProductProjectiveSpaces([P1,P2])
        Traceback (most recent call last):
        ...
        AttributeError: Components must be over the same base ring
    """
    if isinstance(R, (list, tuple)):
        n, R = R, n
    if not isinstance(n, (tuple, list)):
        raise TypeError("Must be a list of dimensions")
    if R is None:
        R = QQ  # default is the rationals
    if isinstance(n[0], ProjectiveSpace_ring):
        #this should be a list of projective spaces
        names = []
        N = []
        R = None
        for PS in n:
            if not isinstance(PS,ProjectiveSpace_ring):
                raise TypeError("Must be a list of Projective Spaces or (dimensions,base ring,names)")
            if R is None:
                R = PS.base_ring()
            elif R != PS.base_ring():
                raise AttributeError("Components must be over the same base ring")
            N.append(PS.dimension_relative())
            names += PS.variable_names()
        X = ProductProjectiveSpaces_ring(N, R, names)
        X._components = n
    else:
        if isinstance(R, (list,tuple)):
           n, R = R, n
        if not isinstance(n,(list,tuple)):
            raise ValueError("Need list or tuple of dimensions")
        if not is_CommutativeRing(R):
            raise ValueError("Must be a commutative ring")
        from sage.structure.parent_gens import normalize_names
        n_vars=sum(d+1 for d in n)
        if isinstance(names, basestring):
            names = normalize_names(n_vars, names)
        else:
            name_list = list(names)
            if len(name_list) == len(n):
                names = []
                for name, dim in zip(name_list, n):
                    names += normalize_names(dim+1, name)
            else:
                n_vars = sum(1+d for d in n)
                names = normalize_names(n_vars, name_list)
        X = ProductProjectiveSpaces_ring(n, R, names)
    return(X)