Galois Theory

Galois Theory

Galois theory is related to group theory, which is a powerful method employed in the analysis of abstract and physical systems that contain symmetry. Group theory plays a critical role in many scientific areas: it is the fundamental basis of the space of quantum mechanics; classical mechanical geometric symmetries are understood using group theory; the development of non-Euclidean geometry by Lagrange in the 19th century is hinged on group theory; the development of algebraic structures of linear and vector spaces is based on group theory; and the analysis and understanding of molecular systems in detail is accomplished only by employing group theory. Galois theory is a generalized field theory that is the mathematical interpretation of group theory. Mathematically, group theory is the basis of real analysis and has had powerful implications in the development of many other areas of mathematics. Evariste Galois, a French mathematician, is attributed with formulating Galois theory and recorded his thoughts on the subject the night before he died in 1832. Galois theory stemmed from an undertaking by Galois for a deeper understanding of the essential conditions an equation must satisfy in order for it to be solvable by radicals.

As the name implies, group theory is concerned with the study of groups. Mathematically, a group is a basic structure of modern algebra and has specific properties that arise in connection with symmetry. A group consists of a set of elements, sometimes referred to as operands, and an operation or binary operator that takes any elements of the set and forms another element of the set under certain conditions. An example of a group would be the set of all numbers, including negative numbers and zero, and the operation of addition. The addition operation would take the sum of any two numbers of the set and form another number of the set. All groups have three properties in common: associativity, identity element, and inverses.

Associativity is a property that assures that the outcome of an operation is the same regardless of the order of operation. So (-3 + 4) +1 = -3 + (4 + 1) no matter which order the operation of addition is carried out.

All groups must also contain a special element called the identity element. It's special because when an operation is performed on the identity element and another element of the set, the operation leaves the other element unchanged. So, for example, in the group previously described, when zero is added to a number on the left of the equality sign or on the right of the equality sign, it leaves the number unchanged: 6 + 0 = 0 + 6. In this group zero is defined as the identity element.

The last property common to all groups is the inverse property. An inverse is an element of the set that, when it is combined with the operation of the group on another element of the set, gives the identity element. In the previous example -6 is called the inverse of 6 because when summed with 6 it yields the identity element 0. This property must exist for all elements of the set in a group. An Abelian group is one that satisfies all these properties and satisfies one additional property. The added property is that for every pair of elements in the group, the operation can be performed in either order and yield the same result. For the group described in the previous example: x + y = y + x must be true for all pairs of elements in that group for that group to be Abelian. If a group contains a finite number of elements then the group is called a finite group and the number of elements is called the order of the group.

Evariste Galois, although only 21 at the time of his death, was probably one of the most important contributors to the development of mathematics. Galois expanded the congruence calculus of Gauss to formulate the notion of a field in group theory. He noted that all that is required for a well-defined field is an Abelian group with a second binary operation that contains a unity. Basically what this means is that the Abelian group minus the identity element must be an Abelian group with respect to the second binary operation. A well-defined field is a necessity when building up the necessary theorems for calculus. The field is the fundamental element to the study of the entire base of mathematics. He recorded all of these thoughts the night before he was killed in a duel in 1832. It wasn't until 1843 that Joseph Liouville announced to the French Academy that he had obtained Galois' papers detailing his ideas of group theory. Eventually, in 1846, Liouville published Galois' papers in his journal, Journal de Mathématiques Pures et Appliquées. In 1870 Camille Jordan, another French mathematician, published the full Galois theory in Traite des Substitutions.

Galois theory depends on the concept of a group since it is the mathematical interpretation of group theory. Galois theory has been employed most often in mathematics in the study of the algebraic solvability of polynomial equations. In fact, the most well-known application of Galois theory is in the proof that radicals cannot solve the general quintic equation with rational coefficients. It is a theory that enables elegant handling of polynomials with minimal algebraic manipulations. Galois, using Galois theory, formulated a method of determining when radicals could solve a general polynomial equation and when they could not. This theory enabled the unification of geometry and algebra. Galois' work contributed to the transition from classical algebra to modern algebra.