Galois Theory

Date: 11/20/96 at 10:10:29
From: Kenneth H. L. Chiu
Subject: Galois Theory

Dear Dr. Math,

I have taken a course in Galois Theory and I found it extremely hard 
to understand.

Could you explain the main idea of the theory, IN SIMPLE TERMS, so 
that even a college student can understand?  (Don't just tell me it 
deals with the insolvability of the quintic.  Please tell me what it 
is all about in the theory content.)

Thanks in advance.

Date: 11/20/96 at 15:14:22
From: Doctor Ceeks
Subject: Re: Galois Theory

Hi,

Try to solve x^2 + 1 = 0.

You find that there are two roots, i and -i.  But what is i, and
what is -i?  You can't tell the difference.  *That* is Galois theory.

---

Okay, that's a little terse, but it does contain the essence of Galois
theory.

To understand Galois theory, you have to be comfortable with vector
spaces and finite group theory. If you are not comfortable with those
concepts, you should learn those first. In this response, I shall
assume you are comfortable with those concepts and I will use them
freely.

Basic Galois theory is the study of field extensions of a given field,
which is typically the field of rational numbers. A field is 
essentially a set of numbers which is rich enough so that you can add, 
subtract, multiply, and divide (by anything nonzero). (These 
operations must satisfy some basic rules, of course, like 
distributivity.)

If F is a field, then a field extension of F is another field E which
contains F.

Notice that E is a vector space over F. (If you check the axioms of a 
vector space, you can prove that E is a vector space over F.) If E is 
a finite dimensional vector space over F, then E is a finite extension 
of F.

Basic Galois theory concerns itself with finite extensions.

It turns out that all finite extensions of F can be obtained by taking 
the smallest field which contains F and the root of some polynomial 
with coefficients in F.

And so, the study of Galois theory is tantamount to the study of the 
roots of polynomials.

What Galois noticed is that such field extensions have a very natural
set of symmetries...sometimes you can interchange roots and induce a
field automorphism.  These symmetries are collected together in the
"Galois Group".

Let us look at the little anecdote I started with. In that case, F is 
the field of real numbers. E is the Complex field. We can obtain E 
by adjoining i = root(-1) to F. Now, when you read the anecdote, you 
may have thought, what do you mean, what is i and what is -i?  i is 
the thing with coordinates (0,1) in the Complex plane, and -i is the 
thing with coordinates (0,-1). But the fact is, it didn't have to be 
that way.  The first person who made the Complex plane could have 
labelled the point (0,1) as -i.  Math would have proceeded just fine.  
You can't tell that there is a symmetry involved.  This symmetry, 
which allows you to interchange i and -i, in fact, is the generator of 
the Galois group in this case.  It is also known as "complex 
conjugation".

Now, group theory is a rich enough subject. There are subgroups, 
normal subgroups, group homomorphisms, centers of groups, etc....
And so naturally, one asks how these various concepts, when applied
to the Galois group, relate to the notion of field extension. The
culminating theorem is known as the fundamental theorem of Galois 
theory, which relates subgroups of the Galois group to intermediary 
field extensions, fields in E which also contain F.

The study of roots of polynomials is also a rich subject. From this
point of view, you get all kinds of terms, like "the splitting field
of a polynomial", which is the smallest field which contains all the
roots of a polynomial...or the "algebraic closure of a field F", which
is the smallest field which contains F and every root of every
polynomial with coefficients in F.

More advanced Galois theory allows for infinite extensions.  Today,
mathematicians are still trying to understanding the nature of the
Galois group of the algebraic closure of the rational numbers over the 
rational numbers.  A lot of progress has been made, but there is much 
to be done.  It's an active area of research, and involves such 
mathematicians as Andrew Wiles (the one who settled the issue of 
Fermat's Last Theorem).

If you find this response unsatisfactory, please write back again
with more specific questions so I can be more accommodating to what
you are after.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/