Permutation Group Approach To Galois Theory
Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it may be that for two of the roots, say A and B, that A2 + 5B3 = 7. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. (One might instead specify a certain field in which the coefficients should lie but, for the simple examples below, we will restrict ourselves to the field of rational numbers.)
These permutations together form a permutation group, also called the Galois group of the polynomial (over the rational numbers). To illustrate this point, consider the following examples:
Read more about this topic: Galois Theory
Famous quotes containing the words group, approach and/or theory:
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)
“Every man alone is sincere. At the entrance of a second person, hypocrisy begins. We parry and fend the approach of our fellow-man by compliments, by gossip, by amusements, by affairs. We cover up our thought from him under a hundred folds.”
—Ralph Waldo Emerson (1803–1882)
“The whole theory of modern education is radically unsound. Fortunately in England, at any rate, education produces no effect whatsoever. If it did, it would prove a serious danger to the upper classes, and probably lead to acts of violence in Grosvenor Square.”
—Oscar Wilde (1854–1900)