Solvability of Polynomials

By William Trad, The University of Sydney

It is well known that quadratic equations have a general solution in terms of the usual operations as well as radicals. This is a fact we all know of since high school. Early mathematicians then asked the question of whether there was an algebraic solution to general cubic polynomials. Such a solution for general cubics was found by Tartaglia in 1530. The next step was to find algebraic solutions to general quartics. Algebraic solutions for general quartics (polynomials of degree 4) were found by Ferrari in 1540. The next logical question is to ask whether the general quintic (polynomials of degree 5) admits an algebraic solution. Unfortunately, the general quintic is unsolvable by radicals (algebraic solutions). There do however exist, quintic polynomials which are solvable by radicals—specifically, reducible quintics. Reducible quintics (quintics that can be broken down into lower order polynomials) can be trivially algebraically solved given the previous progress made by Tartaglia and Ferrari. There do however exist quintics which can’t be broken down however, these can still be solved. These are known as irreducible quintics the question of whether or not a particular irreducible quintic could be solved or not was answered by Arthur Cayley. Arthur Cayley was able to find general criterion for whether an irreducible quintic was solvable algebraically (by radicals) using Galois theory. The foundations of this theory were laid by Evariste Galois. This theory provides a link between two seemingly different fields of mathematics—specifically, field theory and group theory. Typical problems of field theory involve trying to find the minimal “field extension” such that a polynomial is solvable, or compass-straightedge construction problems being solved based upon transcendence or algebraic properties of certain field extensions. Typical problems of Group theory can be thought of as problems of symmetry. Using Galois theory, problems in Field theory can be reduced to problems in Group theory. In the 19th century, Galois theory formed the basis of a proof by Abel and Ruffini to show that there is no algebraic solution (solution in radicals) to general polynomials of degree at least 5.

 

William Trad was a recipient of a 2018/19 AMSI Vacation Research Scholarship.

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