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:
“JuryA group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)
“F.R. Leaviss eat up your broccoli approach to fiction emphasises this junkfood/wholefood dichotomy. If reading a novelfor the eighteenth century reader, the most frivolous of diversionsdid not, by the middle of the twentieth century, make you a better person in some way, then you might as well flush the offending volume down the toilet, which was by far the best place for the undigested excreta of dubious nourishment.”
—Angela Carter (19401992)
“The great tragedy of sciencethe slaying of a beautiful theory by an ugly fact.”
—Thomas Henry Huxley (18251895)