Why is there no nontrivial homomorphism between $\mathbb{Q}[x]/(x^3-1)$ and $\mathbb{Q}[x]/(x^2+x+1)$?
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2026-03-25 16:08:22.1774454902
Nontrivial homomorphism between $\mathbb{Q}[x]/(x^3-1)$ and $\mathbb{Q}[x]/(x^2+x+1)$
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Since $x^3-1=(x-1)(x^2+x+1)$ and the two factors are coprime, we get $$ \mathbb{Q}[x]/(x^3-1) \cong \mathbb{Q}[x]/(x-1) \times \mathbb{Q}[x]/(x^2+x+1) $$ Composing with the projection onto the second factor gives the desired homomorphism.
Actually, this holds more generally. If $g(x)$ divides $f(x)$ in $\mathbb{Q}[x]$, then $$h(x) \bmod (f(x)) \mapsto h(x) \bmod (g(x))$$ is a well-defined homomorphism $\mathbb{Q}[x]/(f(x)) \to \mathbb{Q}[x]/(g(x))$.