Only using integer coefficients, is it possible to find a monic quadratic polynomial that factors normally but doesn’t factor in field/mod 3?
2026-04-13 01:08:22.1776042502
is it possible to find a monic quadratic polynomial that factors in Z but not in Z3?
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No. If we had $$x^2 + Ax + B = (x+a)(x+b)$$ over $\mathbb{Z}$, then we'd have $$x^2+[A]x+[B] = (x+[a])(x+[b])$$ over $\mathbb{Z}/(3\mathbb{Z})$, where $[x] = x+3\mathbb{Z}$ is the quotient map.
This is true just because $x \mapsto x+3\mathbb{Z} : \mathbb{Z} \to \mathbb{Z}/(3\mathbb{Z})$ is a ring homomorphism, and as such, it can generalize to any ring homomorphism (at least for monic polynomials, to avoid one factor becoming a unit).