Galois TheoryDate: 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/ |
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