Online Encyclopedia
Galois theory
In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. In other words, the Galois theory is the study of solutions to polynomials and how the different solutions are related to each other. Symmetries are usually expressed in terms of symmetry groups, and in fact the very notion of a group of permutations was used by Evariste Galois (also Cauchy, and earlier by Ruffini) to describe symmetries of roots. Galois connections describe special relations between partially ordered sets.
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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?"
and
- "Why can't all angles be trisected?"
again meaning, with straight edge and compass. Also it disposed of the question
- "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?"
For details see the following articles:
Symmetries of roots
What exactly do we mean when we say "symmetries of roots of polynomials"? Such a symmetry is a permutation of the roots such that any algebraic equation which is satisfied by the roots is still satisfied after the roots have been permuted. These permutations form a group. Depending on the coefficients we allow in the algebraic equations, one gets thus different Galois groups.
Example of a quadratic equation
As a very simple example, consider any quadratic polynomial
- f(x) = ax^{2}+bx+c.
If we graph y=f(x), we get a parabola with a vertical axis of symmetry across the line x=-b/2a. Reflecting across that line permutes the roots of the polynomial because they are equidistant from the line. Since there are only two roots, it is easy to see that this is the only symmetry of f, so the group of symmetries of f, that is, the Galois group of f, is Z/2Z. Note that the roots may be complex, but they will still be symmetric across x=-b/2a.
Another example
For a trickier example, consider the polynomial
- (x^{2}-5)^{2}-24;
we want to describe the Galois group of this polynomial "over the field of rational numbers" (i.e. allowing only rational numbers as coefficients in the invariant algebraic equations). The roots of the polynomial are
- a = √2 + √3, b = √2 - √3, c = -√2 + √3, d = -√2 - √3.
There are 24 possible ways to permute these four numbers, but not all of them are members of the Galois group. The members of the Galois group must preserve any identity that contains the variables a, b, c and d and rational numbers. One such identity is a + d = 0. Therefore the permutation a→a, b→b, c→d and d→c is not permitted, as a maps to a and d maps to c, but a + c is not zero. A less obvious fact is that (a + b)^{2} = 8. Therefore, we could send (a, b) to (c, d), as we also have (c + d)^{2} = 8, but we could not send (a, b) to (a, c) as (a + c)^{2} = 12. On the other hand, we can send (a, b) to (c, d), despite the fact that a + b = 2√2 and c + d = -2√2. This is because the identity a + b = 2√2 contains an irrational number, and so we don't require the Galois group to preserve it. Putting all this together, we see that the Galois group contains only the following four permutations:
- (a, b, c, d) → (a, b, c, d)
- (a, b, c, d) → (c, d, a, b)
- (a, b, c, d) → (b, a, d, c)
- (a, b, c, d) → (d, c, b, a)
and the Galois group is isomorphic to the Klein four-group.
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 funtion field in one variable over the complex numbers C , and more generally for funtion field 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:
- http://www.math.niu.edu/~beachy/aaol/galois.html
- http://nrich.maths.org/mathsf/journalf/feb02/art2/index_l2h.html
Online textbooks in French, German, Italian and English can be found at:
- http://www.galois-group.net/