Abstract For a Galois extension $K/F$ with $\text {char}(K)\neq 2$ and $\mathrm {Gal}(K/F) \simeq \mathbb {Z}/2\mathbb {Z}\oplus {Z}$ , we determine the $\mathbb {F}_{2}[\mathrm {Gal}(K/F)]$ -module structure of $K^{\times }/K^{\times 2}$ . Although there are an infinite number (pairwise nonisomorphic) indecomposable {F}_{2}[\mathbb {Z}]$ -modules, our decomposition includes at most nine types....

A completely primary finite ring is a ring R with identity 1 = 0 whose subset of all its zero divisors forms the unique maximal ideal J . Let R be a commutative completely primary finite ring with the unique maximal ideal J such that J3 = (0) and J2 = (0). Then R/J ∼= GF(pr) and the characteristic of R is pk, where 1≤ k ≤ 3, for some prime p and positive integer r. Let Ro =GR(pkr , pk) be a Gal...

These notes are based on lectures given by the author at the winter school on Galois theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre’s modularity conjecture and of its proof by Khare, Wintenberger, and Kisin [36] [37] [39], as well as of the results of other mathematicians that played an important role in the proof. Along the way we will r...