perfect field

Noun

 * 1)  A field K such that every irreducible polynomial over K has distinct roots.
 * 2) * 1984, Julio R. Bastida, Field Extensions and Galois Theory,, Addison-Wesley, page 10,
 * If $$K$$ is a perfect field of prime characteristic $$p$$, and if $$n$$ is a nonnegative integer, then the mapping $$\alpha\to\alpha^{p^n}$$ from $$K$$ to $$K$$ is an automorphism.
 * 1) * 2005, Antoine Chambert-Loir, A Field Guide to Algebra, Springer, page 57,
 * Definition 3.1.7. One says a field $$K$$ is perfect if any irreducible polynomial in $$K[X]$$ has as many distinct roots in an algebraic closure as its degree.
 * By the very definition of a perfect field, Theorem 3.1.6 implies that the following properties are equivalent:
 * a) $$K$$ is a perfect field;
 * b) any irreducible polynomial of $$K[X]$$ is separable;
 * c) any element of an algebraic closure of $$K$$ is separable over $$K$$;
 * d) any algebraic extension of $$K$$ is separable;
 * e) for any finite extension $$K\to L$$, the number of $$K$$-homomrphisms from $$K$$ to an algebraically closed extension of $$K$$ is equal to $$[L: K$$].
 * Corollary 3.1.8. Any algebraic extension of a perfect field is again a perfect field.
 * Corollary 3.1.8. Any algebraic extension of a perfect field is again a perfect field.

Usage notes

 * A number of simply stated conditions are equivalent to the above definition:
 * Every irreducible polynomial over $$K$$ is separable;
 * Every finite extension of $$K$$ is separable;
 * Every algebraic extension of $$K$$ is separable;
 * Either $$K$$ has characteristic 0, or, if $$K$$ has characteristic $$p > 0$$, every element of $$K$$ is a $$p$$th power;
 * Either $$K$$ has characteristic 0, or, if $$K$$ has characteristic $$p > 0$$, the Frobenius endomorphism $$x\to x^p$$ is an automorphism of $$K$$;
 * The separable closure of $$K$$ (the unique separable extension that contains all (algebraic) separable extensions of $$K$$) is algebraically closed.
 * Every reduced commutative K-algebra A is a separable algebra (i.e., $$A \otimes_K F$$ is reduced for every field extension $$F/K$$).

Translations

 * French: corps parfait
 * German: perfekte Körper, vollkommene Körper