# source:sage/rings/fraction_field.py@6606:cac6ec5f25c5

Revision 6606:cac6ec5f25c5, 8.7 KB checked in by Mike Hansen <mhansen@…>, 6 years ago (diff)

Changed "Polynomial Ring in ..." to "Multivariate Polynomial Ring in ...".

Line
1"""
2Fraction Field of Integral Domains
3
4AUTHOR: William Stein (with input from David Joyner, David Kohel, and
5        Joe Wetherell)
6
7EXAMPLES:
8Quotienting is a constructor for an element of the fraction field:
9    sage: R.<x> = QQ[]
10    sage: (x^2-1)/(x+1)
11    x - 1
12    sage: parent((x^2-1)/(x+1))
13    Fraction Field of Univariate Polynomial Ring in x over Rational Field
14
15
16The GCD is not taken (since it doesn't converge sometimes) in the inexact case.
17    sage: Z.<z> = CC[]
18    sage: I = CC.gen()
19    sage: (1+I+z)/(z+0.1*I)
20    (1.00000000000000*z + 1.00000000000000 + 1.00000000000000*I)/(1.00000000000000*z + 0.100000000000000*I)
21    sage: (1+I*z)/(z+1.1)
22    (1.00000000000000*I*z + 1.00000000000000)/(1.00000000000000*z + 1.10000000000000)
23
24
25TESTS:
26    sage: F = FractionField(IntegerRing())
28    True
29
30    sage: F = FractionField(PolynomialRing(RationalField(),'x'))
32    True
33
34    sage: F = FractionField(PolynomialRing(IntegerRing(),'x'))
36    True
37
38    sage: F = FractionField(MPolynomialRing(RationalField(),2,'x'))
40    True
41
42"""
43
44#*****************************************************************************
45#
46#   SAGE: System for Algebra and Geometry Experimentation
47#
48#       Copyright (C) 2005 William Stein <wstein@gmail.com>
49#
51#
52#    This code is distributed in the hope that it will be useful,
53#    but WITHOUT ANY WARRANTY; without even the implied warranty of
54#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
55#    General Public License for more details.
56#
57#  The full text of the GPL is available at:
58#
60#*****************************************************************************
61
62import ring
63import integral_domain
64import field
65import fraction_field_element
66import sage.misc.latex as latex
67
68from sage.structure.parent_base import ParentWithBase
69
70def FractionField(R, names=None):
71    """
72    Create the fraction field of the integral domain R.
73
74    INPUT:
75        R -- an integral domain
76        names -- ignored
77
78    EXAMPLES:
79    We create some example fraction fields.
80        sage: FractionField(IntegerRing())
81        Rational Field
82        sage: FractionField(PolynomialRing(RationalField(),'x'))
83        Fraction Field of Univariate Polynomial Ring in x over Rational Field
84        sage: FractionField(PolynomialRing(IntegerRing(),'x'))
85        Fraction Field of Univariate Polynomial Ring in x over Integer Ring
86        sage: FractionField(MPolynomialRing(RationalField(),2,'x'))
87        Fraction Field of Multivariate Polynomial Ring in x0, x1 over Rational Field
88
89    Dividing elements often implicitly creates elements of the fraction field.
90        sage: x = PolynomialRing(RationalField(), 'x').gen()
91        sage: f = x/(x+1)
92        sage: g = x**3/(x+1)
93        sage: f/g
94        1/x^2
95        sage: g/f
96        x^2
97
98    The input must be an integral domain.
99        sage: Frac(Integers(4))
100        Traceback (most recent call last):
101        ...
102        TypeError: R must be an integral domain.
103    """
104    if not ring.is_Ring(R):
105        raise TypeError, "R must be a ring"
106    if not R.is_integral_domain():
107        raise TypeError, "R must be an integral domain."
108    return R.fraction_field()
109
110def is_FractionField(x):
111    return isinstance(x, FractionField_generic)
112
113class FractionField_generic(field.Field):
114    """
115    The fraction field of an integral domain.
116    """
117    def __init__(self, R):
118        """
119        Create the fraction field of the integral domain R.
120
121        INPUT:
122            R -- an integral domain
123
124        EXAMPLES:
125            sage: Frac(QQ['x'])
126            Fraction Field of Univariate Polynomial Ring in x over Rational Field
127            sage: Frac(QQ['x,y']).variable_names()
128            ('x', 'y')
129        """
130        self.__R = R
131        ParentWithBase.__init__(self, R)
132        self._assign_names(R._names)
133
134    def is_field(self):
135        """
136        Returns True, since the fraction field is a field.
137
138        EXAMPLES:
139            sage: Frac(ZZ).is_field()
140            True
141        """
142        return True
143
144    def base_ring(self):
145        """
146        Return the base ring of self; this is the base ring of the ring which
147        this fraction field is the fraction field of.
148
149        EXAMPLES:
150            sage: R = Frac(ZZ['t'])
151            sage: R.base_ring()
152            Integer Ring
153        """
154        return self.__R.base_ring()
155
156    def characteristic(self):
157        """
158        Return the characteristic of this fraction field.
159
160        EXAMPLES:
161            sage: R = Frac(ZZ['t'])
162            sage: R.base_ring()
163            Integer Ring
164            sage: R = Frac(ZZ['t']); R.characteristic()
165            0
166            sage: R = Frac(GF(5)['w']); R.characteristic()
167            5
168        """
169        return self.ring().characteristic()
170
171    def __repr__(self):
172        return "Fraction Field of %s"%self.ring()
173
174    def _latex_(self):
175        return "\\mbox{\\rm Frac}(%s)"%latex.latex(self.ring())
176
177    def ring(self):
178        """
179        Return the ring that this is the fraction field of.
180
181        EXAMPLES:
182            sage: R = Frac(QQ['x,y'])
183            sage: R
184            Fraction Field of Multivariate Polynomial Ring in x, y over Rational Field
185            sage: R.ring()
186            Multivariate Polynomial Ring in x, y over Rational Field
187        """
188        return self.__R
189
190    def __call__(self, x, y=1, coerce=True):
191        if isinstance(x, fraction_field_element.FractionFieldElement):
192            if x.parent() is self:
193                return x
194            elif x.parent() == self:
195                return fraction_field_element.FractionFieldElement(self, x.numerator(), x.denominator())
196            else:
197                R = self.ring()
198                return fraction_field_element.FractionFieldElement(self, R(x.numerator()), R(x.denominator()))
199        if coerce:
200            R = self.ring()
201            x = R(x); y = R(y)
202        return fraction_field_element.FractionFieldElement(self, x, y,
203                                            coerce=False, reduce = self.is_exact())
204
205    def is_exact(self):
206        """
207        EXAMPLES:
208            sage: Z.<z>=CC[]
209            sage: Z.is_exact()
210            False
211        """
212        try:
213            return self.__is_exact
214        except AttributeError:
215            r = self.ring().is_exact()
216            self.__is_exact = r
217        return r
218
219    def construction(self):
220        from sage.categories.pushout import FractionField
221        return FractionField(), self.ring()
222
223    def _coerce_impl(self, x):
224        """
225        Return the canonical coercion of x into this fraction field, or raise a TypeError.
226
227        The rings that canonically coerce to the fraction field are
228           * the fraction field itself
229           * any fraction field that of the form Frac(S) where S canonically
230             coerces to this ring.
231           * any ring that canonically coerces to the ring R such that this
232             fraction field is Frac(R)
233
234        """
235        try:
236            P = x.parent()
237            if is_FractionField(P):
238                R = P.ring()
239                S = self.ring()
240                if S.has_coerce_map_from(R):
241                    return self(x)
242            else:
243                S = self.ring()
244                if S.has_coerce_map_from(P):
245                    return self(x)
246        except AttributeError:
247            pass
248        return self._coerce_try(x, [self.ring()])
249
250    def __cmp__(self, other):
251        if not isinstance(other, FractionField_generic):
252            return cmp(type(self), type(other))
253        return cmp(self.ring(), other.ring())
254
255    def ngens(self):
256        """
257        This is the same as for the parent object.
258
259        EXAMPLES:
260            sage: R = Frac(PolynomialRing(QQ,'z',10)); R
261            Fraction Field of Multivariate Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field
262            sage: R.ngens()
263            10
264        """
265        return self.ring().ngens()
266
267    def gen(self, i=0):
268        """
269        Return the ith generator of self.
270
271        EXAMPLES:
272            sage: R = Frac(PolynomialRing(QQ,'z',10)); R
273            Fraction Field of Multivariate Polynomial Ring in z0, z1, z2, z3, z4, z5, z6, z7, z8, z9 over Rational Field
274            sage: R.0
275            z0
276            sage: R.gen(3)
277            z3
278            sage: R.3
279            z3
280        """
281        x = self.ring().gen(i)
282        one = self.ring()(1)
283        r = fraction_field_element.FractionFieldElement(self, x, one,
284                                                           coerce=False, reduce=False)
285        return r
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