One More Shortcut to Galois Theory

✍ Scribed by A.W.M. Dress

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 359 KB
• Volume: 110
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1995.1005

✦ Synopsis

In this note-following the line of thought introduced into Galois theory by Emil Artin in (or before?) 1948 -a complete proof of the basic structure theorem of that theory is established in altogether less than 50 lines of text (including formulae), using nothing but Dedekind's lemma, one of its elementary and well-known consequences, and two of the most basic facts of (G)-set theory. In the remaining pages, the same direct approach is used to establish for a finite group (G) of automorphisms of a field (L) with fixed field (K) the canonical (anti-)equivalence of the category of finite-dimensional (L)-split (K)-algebras and the category of finite (G)-sets as well as the basic existence theorem of Galois theory, that is, the fact that the number of (K)-algebra automorphisms of a field extension (L=K\left(\alpha_{1}, \ldots, \alpha_{n}\right)) with (#\left{\alpha_{1}, \ldots, \alpha_{n}\right}=n) and (\prod_{i=1}^{n}\left(X-\alpha_{i}\right) \in K[X]) equals the degree ((L: K)=\operatorname{Dim}_{K} L) of this extension and is bounded from above by (n!).. 1995 Academic Press. Inc.