perfect field

perfect field

English

Noun

perfect field (plural perfect fields)

1. (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 ${\\displaystyle K}$ is a perfect field of prime characteristic ${\\displaystyle p}$, and if ${\\displaystyle n}$ is a nonnegative integer, then the mapping ${\\displaystyle \\alpha \\to \\alpha ^{p^{n}}}$ from ${\\displaystyle K}$ to ${\\displaystyle K}$ is an automorphism.

   • 2001, Tsit-Yuen Lam, A First Course in Noncommutative Rings, Springer, 2nd Edition, 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 ${\\displaystyle K}$ is perfect if any irreducible polynomial in ${\\displaystyle K[X]}$ 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) ${\\displaystyle K}$ is a perfect field;

     b) any irreducible polynomial of ${\\displaystyle K[X]}$ is separable;

     c) any element of an algebraic closure of ${\\displaystyle K}$ is separable over ${\\displaystyle K}$;

     d) any algebraic extension of ${\\displaystyle K}$ is separable;

     e) for any finite extension ${\\displaystyle K\\to L}$, the number of ${\\displaystyle K}$-homomrphisms from ${\\displaystyle K}$ to an algebraically closed extension of ${\\displaystyle K}$ is equal to ${\\displaystyle [L:K}$].

     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 ${\\displaystyle K}$ is separable;
  • Every finite extension of ${\\displaystyle K}$ is separable;
  • Every algebraic extension of ${\\displaystyle K}$ is separable;
  • Either ${\\displaystyle K}$ has characteristic 0, or, if ${\\displaystyle K}$ has characteristic ${\\displaystyle p>0}$, every element of ${\\displaystyle K}$ is a ${\\displaystyle p}$th power;
  • Either ${\\displaystyle K}$ has characteristic 0, or, if ${\\displaystyle K}$ has characteristic ${\\displaystyle p>0}$, the Frobenius endomorphism ${\\displaystyle x\\to x^{p}}$ is an automorphism of ${\\displaystyle K}$;
  • The separable closure of ${\\displaystyle K}$ (the unique separable extension that contains all (algebraic) separable extensions of ${\\displaystyle K}$) is algebraically closed.
  • Every reduced commutative K-algebra A is a separable algebra (i.e., ${\\displaystyle A\\otimes _{K}F}$ is reduced for every field extension ${\\displaystyle F/K}$).

Hyponyms

• Galois field

Translations

field such that every irreducible polynomial over it has distinct roots

French: corps parfait m German: perfekte Körper n, vollkommene Körper n

Further reading

• Quasi-finite field on Wikipedia.Wikipedia
• Perfect field on Encyclopedia of Mathematics
• Perfect Field on Wolfram MathWorld