Perfect Field

Perfect Field

In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:

• Every irreducible polynomial over k has distinct roots.
• 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, when k has characteristic p > 0, every element of k is a pth power.
• Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x→xp is an automorphism of k
• The separable closure of k is algebraically closed.
• Every reduced commutative k-algebra A is a separable algebra; i.e., is reduced for every field extension F/k. (see below)

Otherwise, k is called imperfect.

In particular, all fields of characteristic zero and all finite fields are perfect.

Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).

More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism. (This is equivalent to the above condition "every element of k is a pth power" for integral domains.)