Galois Theory Research Papers - Academia.edu (original) (raw)

I Introduction This is written on a Conjecture on Galois Groups for Quadratic Equation. II Conjecture on Galois Groups for Quadratic Equation
In mathematics, more specifically in abstract algebra, Galois theory, named after Evariste Galois,-Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as Which regular polygons are constructible polygons?[1] Why is it not possible to trisect every angle using a compass and straightedge?[1] History Galois theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x-a)(x-b) = x2 + (a + b)x +ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables. Evariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots – in modern terms, its Galois group – had a certain structure-in modern terms, whether or not it was a solvable group. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degree. The Galois proof has three ingredients: The First is: Galois Group is a set of permutations. (p. 170) Mario Livio. The Equation That Couldn't Be Solved, c. 2005. Now examine the quadratic equation. We can denote its two putative^1. solutions by xsub1 and xsub2. Clearly, the combination that is the sum of the two solutions, (xsub1 and xsub2) remains unchanged under the operation of both members of the group of permutations of two objects. The identity leaves xsub1 and xsub2 intact, and exchanging xsub1 and xsub2 simply transforms (xsub1 + xsub2) into (xsub2 + xsub1), which has the same value. For equations of degree n, we know from Gauss's fundamental theorem of algebra that they have n solutions. The maximum number of possible permutations of n solutions is n!, and the group containing all of these permutations is the group we previously called Ssubn. (p. 170) [ It should be obvious for the quadratic equation that 2 solutions provide 2*(2-1) permutations, which are (xsub1 or xsub2) and (xsub1 + xsub2). ] [ mine ] [ The second Galois proof defines a normal subgroup, and a maximal normal