Galois Theory

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Galois theory

[′gal‚wä ‚thē·ə·rē]
The study of the Galois field and Galois group corresponding to a polynomial.

Galois Theory


the theory created by E. Galois of higher degree algebraic equations with one unknown, that is, of equations having the form

(1) xn + a1xn-1 + a2xn-2 + … + + an-1x + an = 0

The theory determines the conditions for reducing the solution of such equations to the solution of a chain of other algebraic equations (usually of lower degree). Inasmuch as the solution of the binomial equation xm = A is the radical Galois Theory, equation (1) is solvable by radicals if it can be reduced to a chain of binomial equations. All equations of degrees two, three, and four are solvable by radicals. The second-degree equation x2 + px + q = 0 was solved in very ancient times by the well-known formula Galois Theory. Third- and fourth-degree equations were solved in the 16th century. For third-degree equations having the form x3 + px + q = 0 (to which all third-degree equations can be reduced), a solution is given by the so-called Cardan’s formula

It was published by J. Cardan in 1545, although the question of whether he discovered it himself or borrowed it from other mathematicians cannot be considered entirely resolved. The method of solving fourth-degree equations by radicals was found by L. Ferrari.

In the course of the next three centuries, mathematicians attempted to find similar formulas for equations of degree five or higher. E. Bezout and J. Lagrange worked persistently on this problem. The latter examined special linear combinations of the roots (the so-called Lagrange resolvents) as well as the question of which equations are satisfied by the rational functions of the roots of equation (1). In 1801, K. Gauss developed a complete theory of a solution by radicals for a binomial equation of the form xn = 1, in which he reduced the solution of such an equation to the solution of a chain of binomial equations of lower degrees and gave the necessary and sufficient conditions for solving the equation xn = 1 in square roots. From a geometric viewpoint, the latter problem consists in finding n-sided regular polygons that can be drawn with a compass and a ruler; consequently the equation xn = 1 is called the cyclotomic equation. Finally, N. Abel demonstrated in 1824 that the general fifth-degree equation (and especially general equations of higher degrees) are not solvable by radicals. On the other hand, Abel gave a solution by radicals for one general class of equations that includes equations of arbitrarily high degrees, the so-called Abelian equations.

Thus, when Galois began his studies much had been accomplished in the theory of algebraic equations but a general theory covering all possible equations of the form of equation (1) had not yet been developed. For instance, it remained (1) to establish the necessary and sufficient conditions that equation (1) must satisfy so that it would be solvable by radicals, (2) to find the chain of simpler equations, even if they were not binomial equations, to which the solution of the given equation (1) could be reduced, and especially (3) to ascertain the necessary and sufficient conditions such that the equation (1) could be reduced to a chain of quadratic equations (that is, so that the roots of the equation could be constructed geometrically with a compass and ruler). Galois answered all these questions in his Memoir on the Conditions of Solvability of Equations by Radicals, which was found among his papers after his death and was first published by J. Liouville in 1846. In order to answer these questions, Galois studied the profound relationships between the properties of equations and groups of substitutions, introducing a number of fundamental concepts of group theory. He formulated his condition of solvability of equation (1) by radicals using group theory terminology. After Galois, his theory was expanded and generalized in many directions. In the modern conception, it is a theory that examines various mathematical entities on the basis of their groups of automorphisms (thus, for example, it is possible to have a Galois theory of fields, of rings, of topological spaces, and so forth).


Galois, E. Sochineniia. Moscow-Leningrad, 1936. (Translated from French.)
Chebotarev, N. G. Osnovy teorii Galua, vols. 1-2. Moscow-Leningrad, 1934-37.
Postnikov, M. M. Teoriia Galua. Moscow, 1963.
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The centerpiece is the development of the Galois theory and its important applications, especially the insolvability of the quintic.
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Galois theory for beginners; a historical perspective.
This comprehensive examination of the theory of valued and ordered fields describes their classical aspects, such as their arithmetic, topology, and Galois theory.
Assuming some knowledge of Galois theory and elementary understanding of point set topology, Artin covers general valuation theory in terms of valuations of a field, complete fields, e, f and n, ramification theory, and the different; in local class field theory he covers the first and second inequalities, the norm residue symbol, the existence theorem, and applications; in product formulae and function fields in one variable he covers preparation for the global theory, characterization of fields by the product formula, differentials in PF-fields, the Reimann-Roch theorem, constant field extensions, applications differentials in function fields and theorems on p-groups and Sylow groups.
Marcolli defines the noncommutative spaces and spectral triples, then describes noncommutable modular curves, quantum statistical mechanics and Galois theory, and noncommutative geometry at arithmetric infinity.