The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in . Then the image of does not contain . If it did, then there would be a rational function whose -th power would equal . But the degree of this -th power is , which is a multiple of . In particular, it can't be 1, which is the degree of . This is a contradiction; so is not in the image of .