perfect field

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perfect field

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Noun

perfect field (plural perfect fields)

(algebra, field theory) A field K such that every irreducible polynomial over K has distinct roots.
- 1984, Julio R. Bastida, Field Extensions and Galois Theory, Cambridge University Press, 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.
- 2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116:
  So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.
- 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.

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 $F/K$ ).