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

[Math Amateur]

Unique Factorization Domain? Nature of Q_Z[x]

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

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

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

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 $$\mathbb{Q}_\mathbb{Z}[x]$$

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

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

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

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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

[This has also been posted on MHF}

 

[jvicens]

Hi Peter,

Your reasoning for (a) is correct. To show that p is a prime in \mathbb{Q}_\mathbb{Z}[x], we can use the fact that if p is a prime in \mathbb{Z}, then it is also a prime in any ring that contains \mathbb{Z}, such as \mathbb{Q}_\mathbb{Z}[x]. This means that if p divides a product of elements in \mathbb{Q}_\mathbb{Z}[x], then it must divide at least one of the elements in the product.

For (b), we can use a similar approach. Suppose p and q are associates in \mathbb{Q}_\mathbb{Z}[x], which means that there exists a unit u in \mathbb{Q}_\mathbb{Z}[x] such that p = uq. Since p and q are positive primes in \mathbb{Z}, this means that u must be a positive rational number. However, this contradicts the fact that p and q are both integers, and therefore cannot be associates in \mathbb{Q}_\mathbb{Z}[x].

Hope this helps!