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Concept# Perfect field

Summary

In algebra, a field k is perfect if any one of the following equivalent conditions holds:
Every irreducible polynomial over k has distinct roots.
Every irreducible polynomial over k is separable.
Every finite extension of k is separable.
Every algebraic extension of k is separable.
Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power.
Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x ↦ x^p is an automorphism of k.
The separable closure of k is algebraically closed.
Every reduced commutative k-algebra A is a separable algebra; i.e., is reduced for every field extension F/k. (see below)
Otherwise, k 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 of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism. (When restricted to integral domains, this is equivalent to the above condition "every element of k is a pth power".)
Examples of perfect fields are:
every field of characteristic zero, so and every finite extension, and ;
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 in characteristic p > 0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is the field , since the Frobenius sends and therefore it is not surjective. It embeds into the perfect field
called its perfection.

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