Perfect field

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

Otherwise, k 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).

More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1] (This is equivalent to the above condition "every element of k is a pth power" for integral domains.)

Examples[edit]

Examples of perfect fields are:

In fact, most fields that appear 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 k(X) of all rational functions in an indeterminate X, where k has characteristic p>0 (because X has no p-th root in k(X)).

Field extension over a perfect field[edit]

Any finitely generated field extension over a perfect field is separably generated.[2]

Perfect closure and perfection[edit]

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 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 A \otimes_k k^{p^{-\infty}} is reduced.[3]

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.[4]

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

\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 x_{i+1}^p=x_i for all i. The map θ : R(A) → A sends (xi) to x0.[5]

See also[edit]

Notes[edit]

  1. ^ Serre 1979, Section II.4
  2. ^ Matsumura, Theorem 26.2
  3. ^ Cohn 2003, Theorem 11.6.10
  4. ^ Bourbaki 2003, Section V.5.1.4, page 111
  5. ^ Brinon & Conrad 2009, section 4.2

References[edit]

External links[edit]