Perfect field

Examples of perfect fields are:

• every field of characteristic zero, so ${\displaystyle \mathbb {Q} }$ and every finite extension, and ${\displaystyle \mathbb {C} }$;[2]
• every finite field ${\displaystyle \mathbb {F} _{q}}$;[3]
• 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 ${\displaystyle \mathbf {F} _{q}(x)}$, since the Frobenius sends ${\displaystyle x\mapsto x^{p}}$ and therefore it is not surjective. It embeds into the perfect field

${\displaystyle \mathbf {F} _{q}(x,x^{1/p},x^{1/p^{2}},\ldots )}$

called its perfection. Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example,[4] consider ${\displaystyle f(x,y)=x^{p}+ay^{p}\in k[x,y]}$ for ${\displaystyle k}$ an imperfect field of characteristic ${\displaystyle p}$ and a not a p-th power in k. Then in its algebraic closure ${\displaystyle k^{\operatorname {alg} }[x,y]}$, the following equality holds:

${\displaystyle f(x,y)=(x+by)^{p},}$

where b^p = a and such b exists in this algebraic closure. Geometrically, this means that ${\displaystyle f}$ does not define an affine plane curve in ${\displaystyle k[x,y]}$.

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 p^r-th roots (r ≥ 1) is perfect; it is called the perfect closure of k and usually denoted by ${\displaystyle k^{p^{-\infty }}}$.

The perfect closure can be used in a test for separability. More precisely, a commutative k-algebra A is separable if and only if ${\displaystyle A\otimes _{k}k^{p^{-\infty }}}$ is reduced.[6]

In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring A_p of characteristic p together with a ring homomorphism u : A → A_p such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : A_p → 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.[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 : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system

${\displaystyle \cdots \rightarrow A\rightarrow A\rightarrow A\rightarrow \cdots }$

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 ${\displaystyle x_{i+1}^{p}=x_{i}}$ for all i. The map θ : R(A) → A sends (x_i) to x₀.[8]

• p-ring
• Perfect ring
• Quasi-finite field

• "Perfect field", Encyclopedia of Mathematics, EMS Press, 2001 [1994]