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Concept of Galois Theory

Evariste Galois was an important and renowned French mathematician of the 18th century (1811-1832). One of his greatest discoveries is the so-called Galois Theory. The story goes that the night before he died he wrote a letter in which he develops the famous theory, with which geometry and algebra are admirably fused. It is a theory that is characterized by the set of results that link the theory of bodies with the theory of groups.

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The origin of Galois Theory was caused by the attempt to answer the absence of a formula that would determine the resolution of equations of polynomials of a fifth or higher degree in terms of the coefficients of the polynomial, with the use of algebraic operations or the root extraction. The aforementioned was possible only for equations of the second degree, third degree and fourth degree.

Often known as a theory for non-mathematicians, Galois then demonstrated almost simultaneously with another genius of mathematical sciences, Niles Henrik Abel, that there is no possibility of finding a general answer for equations of degree 5 only with the use of addition, subtraction, multiplication, division, exponentiation and radication of coefficients (that is, using radicals). It is then concluded that the equations of degree 5 can be solved only with the use of numerical calculation techniques. But there are also many equations of degree 5 or higher, which can be solved correctly by radicals, these would be special cases. Galois formulated and proved a theorem, which is generally called Galoiss theorem. This theorem allows the identification of the aforementioned equations,

If in a polynomial equation the highest power corresponds to a prime number and if you also have knowledge of two values of x, the others can be obtained from them through the use of addition, subtraction, multiplication and division, so the equation can be solved by means of radicals.

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For a form not as complex as the one provided by the previously expressed theorem, we will proceed to identify equations of degree 5 and to equations greater than 5, which can be solved using radicals. It is then necessary to find a new concept, which would be the concept of the group, which is quite complex so we are going to try to introduce it simply.

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First, it is necessary to pay attention to the ordering of the letters or numbers, these are called permutations. The numbers 1, 2 and 3 can be ordered in the following ways 123, 132, 213, 231, 312 and 321. We then call the permutation 123 permutation identity and we will then consider a way of formulating the permutations so that we are left with a pair of lines with the identity on the top line and the corresponding permutation on the bottom line. Thus we have:

Once the above is established, we can define a binary operation on the set of permutations. We will take into account two permutations of any type:

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If we consider first the second permutation and then the first, we can see that what has been done is to relate in sequence the numbers that are derived by the second permutation with those that are derived by the first. This operation. That is, this product is internal since the product of two permutations will result in another permutation. Other properties can also be verified that make the set of permutations conform to a Group structure in proportion to this operation. The properties are:

1. Associative property: The ordering when two contiguous permutations are combined is not of great importance. If we call a, b, and c three permutations and * the operation, the property can be represented in this way, (a * b) * c = a * (b * c).

2. Neutral element: There is a permutation, which can be expressed as e, so that any permutation a, it will be verified that a * e = a. In the case studied, the neutral permutation is the following,

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3. Inverse element: Given any permutation to, there will be another, which will be denoted by,

a^-1+ a l que a*a^-1 = e

For example, if the permutation is considered

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Galoiss investigations have endured in a specific way, coming to formulate a condition so that an equation of polynomials of any type can be determined using radicals.

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