is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect. Fixed points of the Frobenius endomorphism Consider the finite field . By Fermat's little theorem, every element of satisfies . Equivalently, it is a root of the polynomial . The elements of therefore determine roots of this equation, and because this equation has degree it has no more than roots over any extension. In particular, if is an algebraic extension of (such as the algebraic closure or another finite field), then is the fixed field of the Frobenius automorphism of . Let be a ring of characteristic . If is an integral domain, then by the same reasoning, the fixed points of Frobenius are the elements of the prime field. However, if is not a domain, then may have more than roots; for example, this happens if . A similar property is enjoyed on the finite field by the nth iterate of the Frobenius automorphism: Every element of is a root of , so if is an algebraic extension of and is the Frobenius automorphism of , then the fixed field of is . If R is a domain which is an -algebra, then the fixed points of the nth iterate of Frobenius are the elements of the image of . Iterating the Frobenius map gives a sequence of elements in : This sequence of iterates is used in defining the Frobenius closure and the tight closure of an ideal. As a generator of Galois groups The Galois group of an extension of finite fields is generated by an iterate of the Frobenius automorphism. First, consider the case where the ground field is the prime field . Let be the finite field of elements, where . The Frobenius automorphism of fixes the prime field , so it is an element of the Galois group . In fact, since is cyclic with elements, we know that the Galois group is cyclic and is a generator. The order of is because acts on an element by sending it to , and this is the identity on elements of . Every automorphism of is a power of , and the generators are the powers with coprime to . Now consider the finite field as an extension of , where as above. If , then the Frobenius automorphism of does not fix the ground field ,