# Maximal Ideals of Prime and Primitive Polynomials

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Suppose that p is a prime and f is a primitive polynomial in Z[x] which is irreducible modulo p. Show that the set of maximal ideals in Z[x] all can be written as (p,f). Also show that there does not exist a principal ideal in Z[x] which is maximal.

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This solution helps show that a set of maximal ideals in a function can be written in a certain way.

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Let M be a maximal ideal of Z[x].

Let us first show that M∩Z≠{0}. To this end, assume that M∩Z={0} . Let N be the ideal of Q[x ] generated by M. Since Z[x]≠Q[x], N≠Q[x]. Since Q is a field, every proper ideal of Q[x] is principal. Therefore, N=(f(x)) , where f(x)∈Q[x ] and degf(x)>0. Without loss of generality, we may assume that f(x)∈Z[x], and content of f(x) is 1. Let us show that M=(f(x)) in Z[x]. Indeed, consider g(x) from M. Then g(x)∈N, and hence g(x)=f(x)h(x), for some h(x)∈Q[x]. Gauss lemma implies that h(x) ∈Z[x]. This implies that M=(f(x)) in Z[x].

Let us now show that Z[x]/(f(x)) is not a field. Since deg ...

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