Perfect field

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

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).

Another important property of perfect fields is that they admit Witt vectors.

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.)

Examples of perfect fields are:

In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic p>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is

Any finitely generated field extension over a perfect field is separably generated.

One of the equivalent conditions says that, in characteristic p, a field adjoined with all p^r-th roots (r≥1) is perfect; it is called the perfect closure of k and usually denoted by ${\displaystyle k^{p^{-\infty }}}$.

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