Suppose that $K$ is a field, $\mathbb{Q} \subseteq K$ and $[K:\mathbb Q]=n$. If there is $$\Phi:K\to M_2(\mathbb{Q})$$ a ring homomorphim, what are the possible values of $n$?
2026-03-25 03:00:37.1774407637
extension field of $\mathbb{Q}$
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If $K=\Bbb Q(\alpha)$, then $\alpha$ is an eigenvalue of $\Phi(\alpha)$. So $\alpha$ has degree $\le 2$ over $\Bbb Q$ and therefore $n\le2$.