As briefly mentioned in the book, for polynomial equations of degree five or higher we cannot find solutions the same way that we found solutions to the equations of degree at most four.

Implicit in this statement is the notion of `solution’ that we are interested in. From the point of view of Abstract Algebra, the expressions we are looking for are what one usually calls solutions `by radicals,’ although we have not made this too precise yet. Informally, we look for a formula in which we are allowed to use:

The coefficients of the given polynomial, and

complex and real numbers,

and in which we can make use of

the elementary operations , and

radicals—i.e., we can take -th roots of any of the expressions we can obtain, for any positive integer

The amazing result that motivates our work through this course is that, indeed, no such formulas exist for polynomials of degree five or higher.

However, it is a bit misleading to read this as saying that no `formulas’ exist at all. The situation is similar to what mathematicians had to face before having the notion of complex numbers. Then, some equations could not be solved, since their solutions would involve the extraction of square roots of negative numbers. Nowadays, we understand that we can solve those equations, as long as the use of complex numbers is allowed, and we cannot otherwise.

Indeed, although no solutions by radicals are possible for the roots of quintic polynomials, if we allow a larger class of operations to be used, then solutions exist. For example, more general hypergeometric series than -th roots allow us to find the roots of the quintic. Although not nearly as popular now as they once were, hypergeometric series (a particular kind of power series) are still fairly used, for example in partition theory.

There is a well known poster from Wolfram Research explaining how Mathematica can be used to solve the quintic with the help of these functions; their webpage actually is very interesting.

Besides the usefulness of hypergeometric functions, Felix Klein found an approach using the symmetries of the Icosahedron. His book, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, is available online for download from Cornell.

These approaches essentially propagate to higher degree equations, with some new obstacles appearing along the way, but in this sense, one can solve any polynomial equation. The solutions are less satisfactory in that general hypergeometric functions are less well understood and much less intuitive than -th roots. For the purposes of the course, we will concentrate on solutions by radicals, since also these are the ones that lend themselves naturally to algebraic (rather than analytic) study, and the arguments work in much more generality than just over the complex numbers.

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