perfect field

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, 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 ${\displaystyle K}$ 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.

• 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 ${\displaystyle F/K}$).

• Galois field

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