Rings exam problem
Let be a field, consider the ring .
1) Let , prove that there exist and unique such that .
2)Prove that the map defined by is a ring morphism.
3)Find the kernel of (i.e: find generators of it)
4) Find explicitly the image of
5)Prove that is irreducible in . Deduce that the image of is a domain.
1) No clue. Iīm thinking that it has to do with some division but I donīt know what...
2) Ugly computation, I used the identity given in part 1) and I think I pulled it off (if thereīs a simpler way of doing it please let me know).
3) It should be I suppose... Am I right?
4) means something??
5) Maybe I can use that once I proved that is irreducible, but I donīt know how (Eisenstein doesnīt work, I think).
1) Division, you said it! Use the fact that .
is invertible in so such that:
, and i.e. for two unique . (can be written bette, when I write the identity refers to a whose coefficients are in so it's equal to a unique polynomial in and there is no problem, same thing fo and )
2) is quite easy! I don't think you really need 1)
3) Here 1) can be useful. You've guessed it, but how to prove that? Take a assume it belongs to and write it with the identity from 1). Your goal is to prove that
4)Take a . Is in ?
5)Suppose is reducible, i.e. .
Then and since is irreducible in which is factorial we have and for an invertible and a . This makes impossible, contradiction.
Therefore is irreducible in and you go on as you said.