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.

My views:

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).

Thanks, guys!!