Perfect field explained

is perfect if any one of the following equivalent conditions holds:

Every irreducible polynomial over

has no multiple roots in any field extension

L/K

Every irreducible polynomial over

has non-zero formal derivative.

Every irreducible polynomial over

is separable.

Every finite extension of

is separable.

Every algebraic extension of

is separable.

Either

has characteristic 0, or, when

has characteristic

p>0

, every element of

is a

-th power.

Either

has characteristic 0, or, when

has characteristic

p>0

, the Frobenius endomorphism

x\mapstox^p

is an automorphism.

The separable closure of

is algebraically closed.

Every reduced commutative

-algebra

is a separable algebra; i.e.,

A ⊗ _kL

is reduced for every field extension

L/K

.Otherwise,

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

(

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

is a

-th power".

Examples

Examples of perfect fields are:

every field of characteristic zero, so

and its finite extensions, as well as

and

;^[2]

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

F_p(t)

of rational polynomials in the unknown

. This can be seen from the fact that the Frobenius endomorphism sends

x\mapstox^p

and therefore is not surjective. Equivalently, one can show that the polynomial

f(x)=x^p-t

, which is an element of

(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

f(x,y)=x^p+ay^p\ink[x,y]

for

an imperfect field of characteristic

and

not a

-th power in

. Then in its algebraic closure

k^{\operatorname{alg}

}[x,y], the following equality holds:

f(x,y)=(x+by)^p,

where

b^p=a

and such a

exists in this algebraic closure. Geometrically, this means that

does not define an affine plane curve in

k[x,y]

Field extension over a perfect field

over a perfect field

is separably generated, i.e. admits a separating transcendence base, that is, a transcendence base

\Gamma

such that

is separably algebraic over

k(\Gamma)

.^[5]

Perfect closure and perfection

Every field can be embedded in a perfect field: in characteristic

, a field

adjoined with all

p^r

-th roots (

r\geq1

) is perfect; it is called the perfect closure of

and usually denoted by

	p^-infty
k

. For example,

F_q(t)

embeds into

	1/p
F
	q(t,t

	1/p²
,t

,\ldots)

The perfect closure can be used in a test for separability. More precisely, a commutative

-algebra

is separable if and only if

A ⊗ _k

	p^-infty
k

is reduced.

The perfect closure can be defined by a universal property: the perfect closure of a ring

of characteristic

is a perfect ring

A_p

of characteristic

together with a ring homomorphism

u:A\toA_p

such that for any other perfect ring

of characteristic

with a homomorphism

v:A\toB

, there is a unique homomorphism

f:A_p\toB

such that

factors through

(i.e.

v=fu

). The perfect closure always exists; the proof involves "adjoining

-th roots of elements of

", similar to the case of fields.^[6]

The perfection of a ring

of characteristic

is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection

R(A)

is a perfect ring of characteristic

together with a map

\theta:R(A)\toA

such that for any perfect ring

of characteristic

equipped with a map

\phi:B\toA

, there is a unique map

f:B\toR(A)

such that

\phi

factors through

\theta

(i.e.

\phi=\thetaf

). The perfection of

may be constructed as follows. Consider the projective system

… → A → A → A → …

where the transition maps are the Frobenius endomorphism. The inverse limit of this system is

R(A)

and consists of sequences

(x_0,x_1,...)

of elements of

such that

	p=x
x
	i

for all

. The map

\theta:R(A)\toA

sends

(x_i)

x₀

.^[7]

Notes and References

, Section II.4
Examples of fields of characteristic zero include the field of rational numbers, the field of real numbers or the field of complex numbers.
Any finite field of order q may be denoted
F_q

, where q = p for some prime p and positive integer k.
Book: James Milne (mathematician)
. Milne. James. James Milne (mathematician). Elliptic Curves. 6.
Matsumura, Theorem 26.2
, Section V.5.1.4, page 111
, section 4.2

Perfect field explained

Examples

Field extension over a perfect field

Perfect closure and perfection

See also

Notes and References