but its th iterate does. The Galois group is cyclic of order and is generated by . It is the subgroup of generated by . The generators of are the powers where is coprime to . The Frobenius automorphism is not a generator of the absolute Galois group because this Galois group is isomorphic to the profinite integers which are not cyclic. However, because the Frobenius automorphism is a generator of the Galois group of every finite extension of , it is a generator of every finite quotient of the absolute Galois group. Consequently, it is a topological generator in the usual Krull topology on the absolute Galois group. Frobenius for schemes There are several different ways to define the Frobenius morphism for a scheme. The most fundamental is the absolute Frobenius morphism. However, the absolute Frobenius morphism behaves poorly in the relative situation because it pays no attention to the base scheme. There are several different ways of adapting the Frobenius morphism to the relative situation, each of which is useful in certain situations. The absolute Frobenius morphism Suppose that is a scheme of characteristic . Choose an open affine subset of . The ring is an -algebra, so it admits a Frobenius endomorphism. If is an open affine subset of , then by the naturality of Frobenius, the Frobenius morphism on , when restricted to , is the Frobenius morphism on . Consequently, the Frobenius morphism glues to give an endomorphism of . This endomorphism is called the absolute Frobenius morphism of , denoted . By definition, it is a homeomorphism of with itself. The absolute Frobenius morphism is a natural transformation from the identity functor on the category of -schemes to itself. If is an -scheme and the Frobenius morphism of is the identity, then the absolute Frobenius morphism is a morphism of -schemes. In general, however, it is not. For example, consider the ring . Let and both equal with the structure map being the identity. The Frobenius morphism on sends to . It is not a morphism of -algebras. If it were, then multiplying by an element in would commute with applying the Frobenius endomorphism. But this is not true because: The former is the action of in the -algebra structure that begins with, and the latter is the action of induced by Frobenius. Consequently, the Frobenius morphism on is not a morphism of -schemes.