I'm scheduled to see my lecturer about this problem, but he isn't available until next week, and I feel that I am likely to get a clearer explanation here.
This is for a Galois Theory course, and he states the following theorem,
Let be a subring of and fix . Then the smallest subring of that contains both and is equal to
and then he writes this subring as .
Then, he writes an example, , and concludes that
This is fine. The problem is when he introduces subrings of two elements. He claims
I don't understand how he has got to here. The second claim is
Intiutively, I think I have to prove and . The first I think is clear - I can't think of how to prove the second.
Sorry about the length of this question. Any help is much appreciated.