perfect field

plural perfect fields

1. field theoryalgebraA 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 α→α from K to K is an automorphism.

   2 more examples

   So far this stronger conjecture has been proved by Nazarova and Roiter over algebraically closed fields, and subsequently by Ringel over perfect fields.

   2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, 2nd edition, Springer, page 116

   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→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.

   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