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:
“The poet who speaks out of the deepest instincts of man will be heard. The poet who creates a myth beyond the power of man to realize is gagged at the peril of the group that binds him. He is the true revolutionary: he builds a new world.”
—Babette Deutsch (1895–1982)
“So live that when thy summons comes to join
The innumerable caravan that moves
To that mysterious realm, where each shall take
His chamber in the silent halls of death,
Thou go not, like the quarry-slave at night,
Scourged to his dungeon, but, sustained and soothed
By an unfaltering trust, approach thy grave
Like one who wraps the drapery of his couch
About him and lies down to pleasant dreams.”
—William Cullen Bryant (1794–1878)
“The theory of the Communists may be summed up in the single sentence: Abolition of private property.”
—Karl Marx (1818–1883)