if $f(X)$ $\in \mathbb{ Z}[X]$ is irreducible over $\mathbb {Q} $, then $f(X)$ is irreducible over $\mathbb {Z}$.
How proof this , I don't know how start proof this problem Please help me Thanks for all
2026-03-31 02:01:42.1774922502
Proof the Gauss lemma
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I believe it's the other way around, and the statement as you have it would be trivial... that is, it should go : polynomial irreducible over $\mathbb Z \implies $ irreducible over $Q$.
There is a proof on Wikipedia .
This was apparently included in his Disquisitiones Arithmeticae (1801).