Perfect field
In algebra, a field k is perfect if any one of the following equivalent conditions holds:
- Every roots.
- Every irreducible polynomial over k is separable.
- Every finite extension of k is separable.
- Every algebraic extension of k is separable.
- Either k has pth power.
- Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x ↦ xp is an automorphism of k.
- The algebraically closed.
- Every k-algebra A is a separable algebra; i.e., is reduced for every field extension F/k. (see below)
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).
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".)
Examples
Examples of perfect fields are:
- every field of characteristic zero, so and every finite extension, and ;[2]
- every finite field ;[3]
- every algebraically closed field;
- the union of a set of perfect fields totally orderedby extension;
- fields algebraic over a perfect field.
Most fields that are encountered in practice are perfect. The imperfect case arises mainly in
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 for an imperfect field of characteristic 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. Geometrically, this means that does not define an
Field extension over a perfect field
Any
Perfect closure and perfection
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 is reduced.[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.[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
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 for all i. The map θ : R(A) → A sends (xi) to x0.[8]
See also
Notes
- ^ Serre 1979, Section II.4
- complex numbers.
- ^ Any finite field of order q may be denoted , where q = pk for some positive integerk.
- ^ Milne, James. Elliptic Curves (PDF). p. 6.
- ^ Matsumura, Theorem 26.2
- ^ Cohn 2003, Theorem 11.6.10
- ^ Bourbaki 2003, Section V.5.1.4, page 111
- ^ Brinon & Conrad 2009, section 4.2
References
- ISBN 978-3-540-00706-7
- Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05
- Cohn, P.M. (2003), Basic Algebra: Groups, Rings and Fields
- Zbl 0984.00001
- Matsumura, Hideyuki (2003), Commutative ring theory, Translated from the Japanese by M. Reid. Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.)
- MR 0554237
External links
- "Perfect field", Encyclopedia of Mathematics, EMS Press, 2001 [1994]