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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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.

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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]

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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

[This has also been posted on MHF}
 
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