# Irreducible polynomial over $\mathbb{Q}[x]$ has even degree if the sum of two distinct roots lies in $\mathbb{Q}$

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?

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 $$r$$.