Exercise 22.4.21 describes the Frobenius Map, an automorphism of a finite field. If F is a finite field in Sage, then End(F) will create the automorphism group of F, the set of all bijective mappings between the field and itself.
(a) Work Exercise 22.4.21 to gain an understanding of how and why the Frobenius mapping is a field automorphism. (Do not include any of this in your answer to this question, but understand that the following will be much easier if you do this problem first.)
(b) For some small, but not trivial, finite fields locate the Frobenius map in the automorphism group. Small might mean p=2,3,5,7 and 3≤n≤10, with n prime versus composite.
(c) Once you have located the Frobenius map, describe the other automorphisms. In other words, with a bit of investigation, you should find a description of the automorphisms which will allow you to accurately predict the entire automorphism group for a finite field you have not already explored. (Hint: the automorphism group is a group. What if you ``do the operation'' between the Frobenius map and itself? Just what is the operation? Try using Sage's multiplicative notation with the elements of the automorphism group.)
(d) What is the ``structure'' of the automorphism group? What special status does the Frobenius map have in this group?
(e) For any field, the subfield known as the fixed field is an important construction, and will be especially important in the next chapter. Given an automorphism τ of a field E, the subset, K={b∈E∣τ(b)=b}, can be shown to be a subfield of E. It is known as the fixed field of τ in E. For each automorphism of E=GF(36) identify the fixed field of the automorphism. Since we understand the structure of subfields of a finite field, it is enough to just determine the order of the fixed field to be able to identify the subfield precisely.