In algebra, a field k is perfect if any one of the following equivalent conditions holds: Otherwise, k is called imperfect.
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".)
The imperfect case arises mainly in algebraic geometry in characteristic p > 0.
of rational polynomials in an unknown element
Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field.
and a not a p-th power in k. Then in its algebraic closure
, the following equality holds: where bp = a and such b exists in this algebraic closure.
does not define an affine plane curve in
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] One of the equivalent conditions says that, in characteristic p, a field adjoined with all pr-th roots (r ≥ 1) is perfect; it is called the perfect closure of k and usually denoted by
The perfect closure can be used in a test for separability.
More precisely, a commutative k-algebra A is separable if and only if
[6] 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 : A → Ap such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : Ap → B 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.
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 : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf).
Consider the projective system where the transition maps are the Frobenius endomorphism.
The inverse limit of this system is R(A) and consists of sequences (x0, x1, ... ) of elements of A such that