Question

i) Find all the maximal ideals in Z.

ii) Find all the maximal ideals in k[x], where K is a field .

Answer 1

I) Maximal ideals of Z are precisely pZ... Try to prove it... You must know mZ are the ideals of Z... Then pZ will be maximal iff p is prime ... Try it prove this.

ii) K is field . So K[x] is euclidean domain.. So its an PID...

Now in a PID maximal ideals are generated by the irreducible elements.... This is also iff and only if condition ... Try to prove it also .

If you now irreducible elements in PID or K[x] .. You are done those elements will generate maximal ideal .