$\mathbb{Q}(t+t^{-1}) \subseteq \mathbb{Q}(t)$, where $t$ is a variable

Question

Let $\mathbb{Q}(t+t^{-1}) \subseteq \mathbb{Q}(t)$, where $t$ is a variable. Prove or disprove that $\mathbb{Q}(t) / \mathbb{Q}(t+t^{-1})$ is a Galois extension.

Any idea or hint?

Answer 1

HINT: What is the degree of the extension?

ANSWER: Note that $t$ is a root of $$(x-t)(x-t^{-1})=x^2-(t+t^{-1})x+1,$$ so the extension is of degree two and hence Galois.