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:
“Now, honestly: if a large group of ... demonstrators blocked the entrances to St. Patrick’s Cathedral every Sunday for years, making it impossible for worshipers to get inside the church without someone escorting them through screaming crowds, wouldn’t some judge rule that those protesters could keep protesting, but behind police lines and out of the doorways?”
—Anna Quindlen (b. 1953)
“Weaving spiders, come not here;
Hence, you longlegged spinners, hence!
Beetles black approach not near;
Worm nor snail, do no offence.”
—William Shakespeare (1564–1616)
“[Anarchism] is the philosophy of the sovereignty of the individual. It is the theory of social harmony. It is the great, surging, living truth that is reconstructing the world, and that will usher in the Dawn.”
—Emma Goldman (1869–1940)