(a) State the Fundamental Theorem of Galois Theory.

(b) What does it mean for an extension $L$ of $\mathbb{Q}$ to be cyclotomic? Show that a cyclotomic extension $L$ of $\mathbb{Q}$ is a Galois extension and prove that its Galois group is Abelian.

(c) What is the Galois group $G$ of $\mathbb{Q}(\eta)$ over $\mathbb{Q}$, where $\eta$ is a primitive 7 th root of unity? Identify the intermediate subfields $M$, with $\mathbb{Q} \leqslant M \leqslant \mathbb{Q}(\eta)$, in terms of $\eta$, and identify subgroups of $G$ to which they correspond. Justify your answers.