The discussion revolves around proving that every ideal in Z[x] generated by (p, f(x)), where f(x) is an irreducible polynomial in Zp, is maximal. Participants are exploring the properties of ideals and their relationships within the context of polynomial rings.
There is ongoing exploration of the methods to prove the maximality of the ideal. Some participants express uncertainty about the effectiveness of their chosen approaches and seek clarification on specific algebraic manipulations and the implications of certain assumptions.
Participants are grappling with the definitions and properties of units in Z[x] and the implications of irreducibility in Zp. There is a noted lack of consensus on the necessary conditions for the gcd of the polynomials involved.
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.WannaBe22 said: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.WannaBe22 said: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?