Perfect field

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

• Every irreducible polynomial over k has distinct roots.
• Every polynomial over k is separable.
• Every finite extension of k is separable. (This implies that every algebraic extension of k is separable.)
• Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power.
• Every element of k is a qth power. (Here, q is the characteristic exponent, equal to 1 if k has characteristic 0, and equal to p if k has characteristic p > 0).
• The separable closure of k is algebraically closed.
• Every k-algebra A is a separable algebra; i.e., $A \otimes_k F$ is reduced for every field extension F/k.

Otherwise, k is called imperfect.

In particular, every field of characteristic zero and finite fields are perfect.

More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.^[1]

Examples

Examples of perfect fields are: a field of characteristic zero, finite fields, algebraically closed fields, the union of perfect fields, fields algebraic over a perfect field. (In particular, an imperfect field is necessarily transcendental over its prime subfield, which is perfect.) On the other hand, if k has a positive characteristic, then k(X), X indeterminate, is not perfect. In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry.

Perfect closure and perfection

The first condition says that, in characteristic p, a field adjoined with all p-th roots (usually denoted by $k^{p^{-\infty}}$) is perfect; it is called the perfect closure, denoted by k_p. Equivalently, the perfect closure is a maximal purely inseparable subextension. If E / k is a normal finite extension, then $E \simeq k_p \otimes_k k_s$.^[2]

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.^[3]

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

$\cdots\rightarrow A\rightarrow A\rightarrow A\rightarrow\cdots$

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 (x_i) to x₀.^[4]

Notes

1. ^ Serre 1979, Section II.4
2. ^ Cohn, Theorem 11.4.10
3. ^ Bourbaki 2003, Section V.5.1.4, page 111
4. ^ Brinon & Conrad 2009, section 4.2

References

• Bourbaki, Nicholas (2003), Algebra II, Springer, ISBN 978-3-540-00706-7 
• Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory, http://math.stanford.edu/~conrad/papers/notes.pdf, retrieved 2010-02-05 
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, 67 (2 ed.), Springer-Verlag, ISBN 978-0-387-90424-5, MR554237 
• Cohn, P.M. (2003), Basic Algebra: Groups, Rings and Fields