Galois Theory
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Authors
Dababneh, Issa
McCulloch, Ryan
Issue Date
2021-04-09
Type
Other
Language
en_US
Keywords
Galois theory
Alternative Title
Abstract
The proof of the unsolvability of quintic polynomials is usually attributed to Everiste Galois. However, Niels Abel first proved the unsolvability of quintics by drawing from the work of many earlier mathematicians such as Ruffini, Lagrange, Vandermonde, and Newton. Abel’s impossibility proof is very convoluted and involves the nesting of radicals. Galois’ approach is more elegant. Even though the general quintic is unsolvable, there exists quintics that are solvable. The power of Galois theory is its ability to discern which quintics are solvable. We explored modern Galois Theory through the classical lens of solving polynomial equations. Galois Theory is the branch of mathematics which investigates the correspondence between the fields formed by the successive adjunction to a field F of the roots of a given polynomial equation, and the groups consisting of certain permutations on the set of these roots. The theory itself should be distinguished from its historically principal application which is the determination of necessary conditions for the solvability of equations by algebraic operations.