Can someone tell me how can I prove that every ideal in Z[x] generated by (p,f(x)) where f(x) is a polynomial that is irreducible in Zp is maximal??
That's a shame -- it's the way I would have done it, but the method I had in mind requires knowledge of finite fields.The first way doesn't help me at all...
Maybe restating it more algebraically would help? For some other element a, You're looking to show the existence of a linear combination (coefficients in Z[x])t seems like the second way is the one we need to use... But I can't figure out what is the contradiction we get by assuming that there is an element k that exists in J but not in I...How does it help us?
The coefficients of linear combinations come from the ring. So, the element f(x) can be produced by f(x) * 1.2. What is the unit element in Z[x]? If it's the same as the one in Z, So how come it "spans" the whole ring? I mean, if 1 is the unit element in Z[x], so how can we produce every element in Z[x] by multiplications of 1?