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:
“Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.”
—Simone Weil (1910–1943)
“Let me approach at least, and touch thy hand.
[Samson:] Not for thy life, lest fierce remembrance wake
My sudden rage to tear thee joint by joint.
At distance I forgive thee, go with that;
Bewail thy falsehood, and the pious works
It hath brought forth to make thee memorable
Among illustrious women, faithful wives:
Cherish thy hast’n’d widowhood with the gold
Of Matrimonial treason: so farewel.”
—John Milton (1608–1674)
“[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)