Perfect field

Examples of perfect fields are:

• every field of characteristic zero, so ${\displaystyle \mathbb {Q} }$ and its finite extensions, as well as ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle p>0}$. 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 ${\displaystyle \mathbb {F} _{p}(t)}$ of rational polynomials in the unknown ${\displaystyle t}$. This can be seen from the fact that the Frobenius endomorphism sends ${\displaystyle x\mapsto x^{p}}$ and therefore is not surjective. Equivalently, one can show that the polynomial ${\displaystyle f(x)=x^{p}-t}$, which is an element of ${\displaystyle (\mathbb {F} _{p}(t))[x]}$, 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,^[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 ${\displaystyle a}$ not a ${\displaystyle p}$-th power in ${\displaystyle k}$. Then in its algebraic closure ${\displaystyle k^{\operatorname {alg} }[x,y]}$, the following equality holds:

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

where ${\displaystyle b^{p}=a}$ and such a ${\displaystyle b}$ 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 ${\displaystyle K}$ over a perfect field ${\displaystyle k}$ is separably generated, i.e. admits a separating transcendence base, that is, a transcendence base ${\displaystyle \Gamma }$ such that ${\displaystyle K}$ is separably algebraic over ${\displaystyle k(\Gamma )}$.^[5]

Every field can be embedded in a perfect field: in characteristic ${\displaystyle p}$, a field ${\displaystyle k}$ adjoined with all ${\displaystyle p^{r}}$-th roots (${\displaystyle r\geq 1}$) is perfect; it is called the perfect closure of ${\displaystyle k}$ and usually denoted by ${\displaystyle k^{p^{-\infty }}}$. For example, ${\displaystyle \mathbb {F} _{q}(t)}$ embeds into ${\displaystyle \mathbb {F} _{q}(t,t^{1/p},t^{1/p^{2}},\ldots )}$.

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

The perfect closure can be defined by a universal property: the perfect closure of a ring ${\displaystyle A}$ of characteristic ${\displaystyle p}$ is a perfect ring ${\displaystyle A_{p}}$ of characteristic ${\displaystyle p}$ together with a ring homomorphism ${\displaystyle u:A\to A_{p}}$ such that for any other perfect ring ${\displaystyle B}$ of characteristic ${\displaystyle p}$ with a homomorphism ${\displaystyle v:A\to B}$, there is a unique homomorphism ${\displaystyle f:A_{p}\to B}$ such that ${\displaystyle v}$ factors through ${\displaystyle u}$ (i.e. ${\displaystyle v=fu}$). The perfect closure always exists; the proof involves "adjoining ${\displaystyle p}$-th roots of elements of ${\displaystyle A}$", similar to the case of fields.^[7]

The perfection of a ring ${\displaystyle A}$ of characteristic ${\displaystyle p}$ is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection ${\displaystyle R(A)}$ of ${\displaystyle A}$ is a perfect ring of characteristic ${\displaystyle p}$ together with a map ${\displaystyle \theta :R(A)\to A}$ such that for any perfect ring ${\displaystyle B}$ of characteristic ${\displaystyle p}$ equipped with a map ${\displaystyle \phi :B\to A}$, there is a unique map ${\displaystyle f:B\to R(A)}$ such that ${\displaystyle \phi }$ factors through ${\displaystyle \theta }$ (i.e. ${\displaystyle \phi =\theta f}$). The perfection of ${\displaystyle A}$ 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 ${\displaystyle R(A)}$ and consists of sequences ${\displaystyle (x_{0},x_{1},\dots )}$ of elements of ${\displaystyle A}$ such that ${\displaystyle x_{i+1}^{p}=x_{i}}$ for all ${\displaystyle i}$. The map ${\displaystyle \theta :R(A)\to A}$ sends ${\displaystyle (x_{i})}$ to ${\displaystyle x_{0}}$.^[8]

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