The question is as in the title:
Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
2026-04-01 14:22:37.1775053357
Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
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This is not true for polynomials of degree $0$. The polynomial $7$ is a prime element in $\mathbb{Z}[X]$ yet not in $\mathbb{Q}[X]$ (as there it is a unit). This works of course for any prime element of $\mathbb{Z}$, that is plus/minus the prime numbers.
For polynomials of positive degree it is true by Gauss's lemma as remarked in a comment. (Also note that the converse, that may seem obvious, is not strictly true for reasons similar to the one mentioned at the start.)