One More Shortcut to Galois Theory

More Notes on Galois Theory

Despite its name, the Fundamental Theorem of Algebra cannot be a result in pure algebra since the real numbers and hence the complex numbers are not algebraically defined. While there are many proofs, most use some basic facts in complex analysis or plane topology. We describe here a proof based on Galois theory as well as some non-trivial finite group theory, namely the Sylow theorems, but whi...

Galois Corings Applied to Partial Galois Theory

Partial Galois extensions were recently introduced by Doku-chaev, Ferrero and Paques. We introduce partial Galois extensions for noncommutative rings, using the theory of Galois corings. We associate a Morita context to a partial action on a ring.

From Galois to Hopf Galois: Theory and Practice

Hopf Galois theory expands the classical Galois theory by considering the Galois property in terms of the action of the group algebra k[G] on K/k and then replacing it by the action of a Hopf algebra. We review the case of separable extensions where the Hopf Galois property admits a group-theoretical formulation suitable for counting and classifying, and also to perform explicit computations an...

Galois Theory

Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...