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Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial and suppose that $\alpha, \beta \in E/\mathbb{Q}$ are two distinct roots of $f(x)$ in its splitting field satisfying $\alpha + \beta \in \mathbb{Q}$.

How do I show that the degree of $f(x)$ is even?

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Suppose $\alpha+\beta = r$.

If $\gamma$ is any root of $f(X)$, pick some automorphism $\tau_\gamma$ of the splitting field sending $\alpha$ to $\gamma$. Then $\gamma + \tau_\gamma(\beta) = r$.

Now partition the roots of $f$ into pairs that add to $r$.

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