def __init__(self,
R,
element_class=fraction_field_element.FractionFieldElement,
category=QuotientFields()):
"""
Create the fraction field of the integral domain R.
INPUT:
- ``R`` - an integral domain
EXAMPLES::
sage: Frac(QQ['x'])
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: Frac(QQ['x,y']).variable_names()
('x', 'y')
sage: category(Frac(QQ['x']))
Category of quotient fields
"""
self._R = R
self._element_class = element_class
self._element_init_pass_parent = False
Parent.__init__(self, base=R, names=R._names, category=category)
def __init__(self):
r"""
We create the rational numbers `\QQ`, and call a few functions::
sage: Q = RationalField(); Q
Rational Field
sage: Q.characteristic()
0
sage: Q.is_field()
True
sage: Q.category()
Join of Category of number fields
and Category of quotient fields
and Category of metric spaces
sage: Q.zeta()
-1
We next illustrate arithmetic in `\QQ`.
::
sage: Q('49/7')
7
sage: type(Q('49/7'))
<type 'sage.rings.rational.Rational'>
sage: a = Q('19/374'); a
19/374
sage: b = Q('17/371'); b
17/371
sage: a + b
13407/138754
sage: b + a
13407/138754
sage: a * b
19/8162
sage: b * a
19/8162
sage: a - b
691/138754
sage: b - a
-691/138754
sage: a / b
7049/6358
sage: b / a
6358/7049
sage: b < a
True
sage: a < b
False
Next finally illustrate arithmetic with automatic coercion. The
types that coerce into the rational field include ``str, int,
long, Integer``.
::
sage: a + Q('17/371')
13407/138754
sage: a * 374
19
sage: 374 * a
19
sage: a/19
1/374
sage: a + 1
393/374
TESTS::
sage: TestSuite(QQ).run()
sage: QQ.variable_name()
'x'
sage: QQ.variable_names()
('x',)
sage: QQ._element_constructor_((2, 3))
2/3
sage: QQ.is_finite()
False
sage: QQ.is_field()
True
"""
from sage.categories.basic import QuotientFields
from sage.categories.number_fields import NumberFields
ParentWithGens.__init__(
self, self, category=[QuotientFields().Metric(),
NumberFields()])
self._assign_names(('x', ), normalize=False) # ?????
self._populate_coercion_lists_(init_no_parent=True)