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A solvable quintic is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals. To characterize solvable quintics, and more generally solvable polynomials of higher degree, Évariste Galois developed techniques which gave rise to group theory and Galois theory.
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) = x 2 – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.
This finishes the proof that the Galois group of a general equation is the symmetric group, and thus proves the original Abel–Ruffini theorem, which asserts that the general polynomial equation of degree n cannot be solved in radicals for n > 4.
The Galois group of a polynomial of degree is or a proper subgroup of it. If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup. Transitive subgroups of form a directed graph: one group can be a subgroup of several groups. One resolvent can tell if the Galois group of a polynomial is a (not ...
One of the basic propositions required for completely determining the Galois groups [3] of a finite field extension is the following: Given a polynomial () [], let / be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,
Its Galois group = (/) comprises the automorphisms of K which fix a. Such automorphisms must send √ 2 to √ 2 or – √ 2, and send √ 3 to √ 3 or – √ 3, since they permute the roots of any irreducible polynomial.
Évariste Galois (/ ɡ æ l ˈ w ɑː /; [1] French: [evaʁist ɡalwa]; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a problem that had been open for 350 years.
The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. [a] A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension. [2]