0Introduction
II Galois Theory
0 Introduction
The most famous result of Galois theory is that there is no general solution
to polynomial equations of degree 5 or above in terms of radicals. However,
this result was, in fact, proven before Galois theory existed, and goes under the
name of the Abel–Ruffini theorem. What Galois theory does provides is a way
to decide whether a given polynomial has a solution in terms of radicals, as well
as a nice way to prove this result.
However, Galois theory is more than equation solving. In fact, the funda-
mental theorem of Galois theory, which is obviously an important theorem in
Galois theory, has completely nothing to do with equation solving. Instead, it is
about group theory.
In modern days, Galois theory is often said to be the study of field extensions.
The idea is that we have a field
K
, and then add more elements to get a field
L
.
When we want to study solutions to polynomial equations, what we add is the
roots of the polynomials. We then study the properties of this field extension,
and in some cases, show that this field extension cannot be obtained by just
adding radicals.
For certain “nice” field extensions
K ⊆ L
, we can assign to it the Galois group
Gal
(
L/K
). In general, given any group
G
, we can find subgroups of
G
. On the
other hand, given a field extension
K ⊆ L
, we can try to find some intermediate
field
F
that can be fitted into
K ⊆ F ⊆ L
. The key idea of Galois theory is
that these two processes are closely related — we can establish a one-to-one
correspondence between the subgroups of G and the intermediate fields F .
Moreover, many properties of (intermediate) field extensions correspond to
analogous ideas in group theory. For example, we have the notion of normal
subgroups, and hence there is an analogous notion of normal extensions. Similarly,
we have soluble extensions (i.e. extensions that can be obtained by adding
radicals), and these correspond to “soluble groups”. In Galois theory, we will
study how group-theoretic notions and field-theoretic notions interact.
Nowadays, Galois theory is an important field in mathematics, and finds its
applications in number theory, algebraic geometry and even cryptography.