Fields

class sage.categories.fields.Fields(s=None)

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()
class ElementMethods
class Fields.ParentMethods
is_integrally_closed()

Return True, as per IntegralDomain.is_integraly_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
Fields.super_categories(*args, **kwds)

EXAMPLES:

sage: Fields().super_categories()
[Category of euclidean domains, Category of unique factorization domains, Category of division rings]

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