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).