| Among problems in mathematics whose search for solutions dominated the lives of renowned mathematicians were related to possible constructions in geometry that could be made by the use of a pair of compasses and a ruler only. They included trisecting any angle, duplicating a cube, squaring a circle and constructing regular polygons for a given number of sides. As progress was made it was realized that such problems were related to the solution of the polynomial of the form f(x) = a0 + a1x + a2x2 +...+ anxn.; Having obtained formulas for solving quadratic, cubic and quartic polynomials, the pattern encouraged them to look for a general formula for solving quintic polynomials and those of a higher degree. Galois considered the solution of the above polynomial in the form of its coefficients a 0, a1,...,a n-1 by using a finite number of operations to obtain field extensions of the coefficient field Q (a0, a1,..., an-1).; In the main theorem of Galois theory he associated such field extensions to a corresponding Galois group and showed the importance of the symmetry and correspondence between them to the solution of polynomials. Galois theory has applications in other areas of mathematics including a relation to the impossibility of the geometric constructions given above. |
