Galois theory

Application to classical problems

Galois theory has applications to classic mathematical problems. These include definitive answers to the geometric questions

"Which regular polygons can be constructed in the classical way with straightedge and compass?"

"Why can't all angles be trisected?"

"Why isn't there a formula for the roots of a 5th or higher degree polynomial in terms of the usual algebraic operations (+,-,×,÷) and the taking of n-th roots?"

• constructible polygon
• trisection of an angle
• Abel-Ruffini theorem

  Symmetries of roots

Example of a quadratic equation

As a very simple example, consider any quadratic polynomial

f(x) = ax²+bx+c.

Another example

For a trickier example, consider the polynomial

(x²-5)²-24;

Modern approach by field theory

In the modern approach, the setting is changed somewhat, in order to achieve a precise and more general definition: one starts with a field extension L/K and defines its Galois group as the group of all field automorphisms of L which keep all elements of K fixed. In the example above, we computed the Galois group of the field extension Q(a,b,c,d)/Q.

Solvable groups and solution by radicals

The notion of a solvable group in group theory allows us to determine whether or not a polynomial is solvable in the radicals, depending on whether or not its Galois group has the property of solvability. In essence, each field extension L/K corresponds to a factor group in a composition series of the Galois group. If a factor group in the composition series is cyclic of order n, then the corresponding field extension is a radical extension, and the elements of L can then be expressed using the nth root of some element of K.

If all the factor groups in its composition series are cyclic, the Galois group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q).

One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals—the Abel-Ruffini theorem. This is due to the fact that for n > 4 the symmetric group S_n contains a simple, non-cyclic, normal subgroup.

Inverse problems

It is easy to construct field extensions with a given finite group as Galois group: Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {x_α}, one for each element α of G, and adjoin them to K to get the field K({x_α}). Call it F. Contained within F is the field L of symmetric rational functions in the {x_α}. The Galois group of L over F is S, by a basic result of Artin. G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the Fundamental Theorem of Galois Theory, the Galois group of F over M is G.

It is an open problem (in general) how to construct field extensions of a fixed ground field with a given finite group as Galois group. This problem was first posed by Emmy Noether for field of rational numbers Q. It is called the inverse Galois problem. There is a great deal of detailed information in particular cases. The problem is solved for function fields in one variable over the complex numbers C, and more generally for function fields in one variable over an algebraically closed field of characteristic 0. Shafarevich solved the problem for finite solvable groups in the case of Q.

External links

Some on-line tutorials on Galois theory appear at: