Perfect field

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