Wikipedia / Perfect field
Wikipedia / Perfect field
In algebra, a field is perfect if any one of the following equivalent conditions holds:
Every irreducible polynomial has no multiple roots in any field extension.
Every irreducible polynomial has non-zero formal derivative.
Every irreducible polynomial is separable.
Every finite extension is separable.
Every algebraic extension is separable.
has characteristic-th power.
has characteristic, the Frobenius endomorphism is an automorphism.
Every reduced-algebra is a separable algebra is reduced for every field extension.
is called imperfect.
In particular, all fields of characteristic zero and all finite fields are perfect.
Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).
Another important property of perfect fields is that they admit Witt vectors.
More generally, a ring a prime) is called perfect if the Frobenius endomorphism is an automorphism. When restricted to integral domains-th power".
Examples
Examples of perfect fields are:
;
every finite field;
every algebraically closed field;
the union of a set of perfect fields totally ordered by extension;
fields algebraic over a perfect field.
Most fields that are encountered in practice are perfect. The imperfect case arises mainly in algebraic geometry. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect.
, is irreducible but inseparable.
Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example,, the following equality holds:
does not define an affine plane curve.
Field extension over a perfect field
Any finitely generated field extension is separably generated, i.e. admits a separating transcendence base.
Perfect closure and perfection
) is perfect; it is called the perfect closure.
is reduced.
The perfect closure can be defined by a universal property together with a ring homomorphism", similar to the case of fields.
The perfection is the dual may be constructed as follows. Consider the projective system
where the transition maps are the Frobenius endomorphism. The inverse limit.
See also
Notes
References
External links