Bases: sage.categories.category.Category
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 unique factorization 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 True, as per IntegralDomain.is_integraly_closed():
for every field ,
is its own field of fractions,
hence every element of
is integral over
.
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
EXAMPLES:
sage: Fields().super_categories()
[Category of euclidean domains, Category of unique factorization domains, Category of division rings]