Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton
The category of (commutative) fields, i.e. commutative rings where all non-zero elements have multiplicative inverses
EXAMPLES:
sage: K = Fields()
sage: K
Category of fields
sage: Fields().super_categories()
[Category of euclidean domains, Category of division rings]
sage: K(IntegerRing())
Rational Field
sage: K(PolynomialRing(GF(3), 'x'))
Fraction Field of Univariate Polynomial Ring in x over
Finite Field of size 3
sage: K(RealField())
Real Field with 53 bits of precision
TESTS:
sage: TestSuite(Fields()).run()
Return the degree of this element as an element of a euclidean domain.
In a field, this returns 0 for all but the zero element (for which it is undefined).
EXAMPLES:
sage: QQ.one().euclidean_degree()
0
Greatest common divisor.
NOTE:
Since we are in a field and the greatest common divisor is only determined up to a unit, it is correct to either return zero or one. Note that fraction fields of unique factorization domains provide a more sophisticated gcd.
EXAMPLES:
sage: GF(5)(1).gcd(GF(5)(1))
1
sage: GF(5)(1).gcd(GF(5)(0))
1
sage: GF(5)(0).gcd(GF(5)(0))
0
For fields of characteristic zero (i.e., containing the integers as a sub-ring), evaluation in the integer ring is attempted. This is for backwards compatibility:
sage: gcd(6.0,8); gcd(6.0,8).parent()
2
Integer Ring
If this fails, we resort to the default we see above:
sage: gcd(6.0*CC.0,8*CC.0); gcd(6.0*CC.0,8*CC.0).parent()
1.00000000000000
Complex Field with 53 bits of precision
AUTHOR:
Returns True if self has a multiplicative inverse.
EXAMPLES:
sage: QQ(2).is_unit()
True
sage: QQ(0).is_unit()
False
Least common multiple.
NOTE:
Since we are in a field and the least common multiple is only determined up to a unit, it is correct to either return zero or one. Note that fraction fields of unique factorization domains provide a more sophisticated lcm.
EXAMPLES:
sage: GF(2)(1).lcm(GF(2)(0))
0
sage: GF(2)(1).lcm(GF(2)(1))
1
If the field contains the integer ring, it is first attempted to compute the gcd there:
sage: lcm(15.0,12.0); lcm(15.0,12.0).parent()
60
Integer Ring
If this fails, we resort to the default we see above:
sage: lcm(6.0*CC.0,8*CC.0); lcm(6.0*CC.0,8*CC.0).parent()
1.00000000000000
Complex Field with 53 bits of precision
sage: lcm(15.2,12.0)
1.00000000000000
AUTHOR:
Return the quotient with remainder of the division of this element by other.
INPUT:
EXAMPLES:
sage: f,g = QQ(1), QQ(2)
sage: f.quo_rem(g)
(1/2, 0)
Compute the extended gcd of self and other.
INPUT:
OUTPUT:
A tuple (r, s, t) of elements in the parent of self such that r = s * self + t * other. Since the computations are done over a field, r is zero if self and other are zero, and one otherwise.
AUTHORS:
EXAMPLES:
sage: (1/2).xgcd(2)
(1, 2, 0)
sage: (0/2).xgcd(2)
(1, 0, 1/2)
sage: (0/2).xgcd(0)
(0, 0, 0)
alias of FiniteFields
Returns the fraction field of self, which is self.
EXAMPLES:
sage: QQ.fraction_field() is QQ
True
Returns True as self is a field.
EXAMPLES:
sage: QQ.is_field()
True
sage: Parent(QQ,category=Fields()).is_field()
True
Return True, as per IntegralDomain.is_integrally_closed(): for every field \(F\), \(F\) is its own field of fractions, hence every element of \(F\) is integral over \(F\).
EXAMPLES:
sage: QQ.is_integrally_closed()
True
sage: QQbar.is_integrally_closed()
True
sage: Z5 = GF(5); Z5
Finite Field of size 5
sage: Z5.is_integrally_closed()
True
Return whether this field is perfect, i.e., its characteristic is \(p=0\) or every element has a \(p\)-th root.
EXAMPLES:
sage: QQ.is_perfect()
True
sage: GF(2).is_perfect()
True
sage: FunctionField(GF(2), 'x').is_perfect()
False
EXAMPLES:
sage: Fields().extra_super_categories()
[Category of euclidean domains]