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In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients.
Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equations. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable [1] (note this theorem holds only in characteristic 0).
According to Nathan Jacobson, "The proofs of Ruffini and of Abel [...] were soon superseded by the crowning achievement of this line of research: Galois' discoveries in the theory of equations." [16] In 1830, Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals, which was ...
The proof is based on the fundamental theorem of Galois theory and the following theorem. Let K be a field containing n distinct n th roots of unity. An extension of K of degree n is a radical extension generated by an nth root of an element of K if and only if it is a Galois extension whose Galois group is a cyclic group of order n.
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.
It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension E / F that is finite and Galois , there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group .
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. Applying these techniques, Arthur Cayley found a general criterion for determining whether any given quintic is solvable. [2] This criterion is the following. [3]
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory , so named in honor of ...