Perfect field

In algebra, a field k is perfect if any one of the following equivalent conditions holds:

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.[1] (When restricted to integral domains, this is equivalent to the above condition "every element of k is a pth power".)

Any finitely generated field extension K over a perfect field k is separably generated, i.e. admits a separating transcendence base, that is, a transcendence base Γ such that K is separably algebraic over k(Γ).[5]

In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring Ap of characteristic p together with a ring homomorphism u : AAp such that for any other perfect ring B of characteristic p with a homomorphism v : AB there is a unique homomorphism f : ApB such that v factors through u (i.e. v = fu). The perfect closure always exists; the proof involves "adjoining p-th roots of elements of A", similar to the case of fields.[7]

The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : BA, there is a unique map f : BR(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system