Let $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$ denote the set of polynomials with rational coefficients and integer constant terms.

(a) If p is prime in $\displaystyle \mathbb{Z} $, prove that the constant polynomial p is irreducible in $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$.

(b) If p and q are positive primes in $\displaystyle \mathbb{Z} $, prove that p and q are not associates in $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$

I am unsure of my thinking on these problems.

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Regarding (a) I think the solution is as follows:

We need to show the p is irreducible in $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$

That is if p = ab for $\displaystyle p, a, b \in \mathbb{Q}_\mathbb{Z}[x] $ then at least one of a or b must be a unit

But then we must have p = 1.p = p.1 since p is a prime in $\displaystyle \mathbb{Z} $ and hence is prime in $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$

But 1 is a unit in $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$ (and also in $\displaystyle \mathbb{Z} $)

Thus p is irreducible in $\displaystyle \mathbb{Q}_\mathbb{Z}[x]$

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Could someone please either confirm that my working is correct in (a) or let me know if my reasoning is incorrect or lacking in rigour.

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Help with the general approach for (b) would be appreciated

Peter