Galois theory Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable of degree , in which the coefficients are drawn from some ground field, which might be, for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of , which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. For example, if the polynomial is and the field is the real numbers, then the polynomial has no roots, because any choice of makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.
Galois theory Galois theory concerns transformations of number fields that permute the roots of an equation. Consider a polynomial equation of a variable of degree , in which the coefficients are drawn from some ground field, which might be for example, the field of real numbers, rational numbers, or the integers modulo 7. There may or may not be choices of , which make this polynomial evaluate to zero. Such choices, if they exist, are called roots. For example, if the polynomial is and the field is the real numbers, then the polynomial has no roots, because any choice of makes the polynomial greater than or equal to one. If the field is extended, however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.