Perfect field

English

Noun

1. Template:Lb A field K such that every irreducible polynomial over K has distinct roots.
   • 1984, Julio R. Bastida, Field Extensions and Galois Theory, Template:W, 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.

   • Template:Quote-book
   • 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 $F/K$).

Hyponyms

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Translations

• French: Template:T
• German: Template:T, Template:T

Further reading

• Template:Pedia
• Perfect field on Template:W
• Perfect Field on Template:W