Galois Theory, 5th Edition

Chapter 16Abstract Rings and Fields

DOI: 10.1201/9781003213949-16

Having seen how Galois Theory works in the context assumed by its inventor, we can generalise everything to a much broader context. Instead of subfields of $ℂ$, we can consider arbitrary fields. This step goes back to Weber in 1895, but first achieved prominence in the work of Emil Artin in lectures of 1926, later published as Artin (1948). With the increased generality, new phenomena arise, and these must be dealt with.

One such phenomenon relates to the Fundamental Theorem of Algebra, which does not hold in an arbitrary field. We can get round this by constructing an analogue, the ‘algebraic closure’ of a field, in which every polynomial splits into linear factors. However, ...