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Unique Factorization Domain? Nature of Q_Z[x] - 1


Well-known member
MHB Site Helper
Jun 22, 2012
Unique Factorization Domain? Nature of Q_Z[x]

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

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

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

I am unsure of my thinking on these problems.


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

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

That is if p = ab for [TEX] p, a, b \in \mathbb{Q}_\mathbb{Z}[x] [/TEX] 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 [TEX] \mathbb{Z} [/TEX] - BUT is it prime in [TEX] \mathbb{Q}_\mathbb{Z}[x][/TEX] (can someone help here???)

But 1 is a unit in [TEX] \mathbb{Q}_\mathbb{Z}[x][/TEX] (and also in [TEX] \mathbb{Z} [/TEX]) - I have yet to properly establish this!)

Thus p is irreducible in [TEX] \mathbb{Q}_\mathbb{Z}[x][/TEX]


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.


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


[This has also been posted on MHF}
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