On the Impossibility of Squaring the Circle

A polynomial is a formal expression of the form \(a_n X^n + a_{n - 1} X^{n - 1} + \cdots + a_1 X + a_0\) in an indeterminate X. Polynomials are fundamental objects of algebra, and we can calculate with them as with numbers. In particular, we can add and multiply polynomials. Just as we can assign the absolute value to each integer as a measure of its size, we can assign the degree to polynomials as a measure. This finds application in division with remainder. If the dividend and divisor are polynomials, we find a quotient so that the remainder has a smaller degree than the divisor. The divisibility theory of integers is also mirrored by polynomials. To what extent a polynomial can be factorized into polynomials of smaller degrees depends crucially on the chosen coefficient ring. We show that the fundamental theorem of algebra implies that polynomials over the algebraic numbers always decompose into a product of linear terms. At the end of this chapter, we apply the results obtained about polynomials to show with an elementary proof that \(\uppi \) is transcendental, i.e., it is not the root of a polynomial equation with rational coefficients. We infer from this that we cannot construct the lengths \(\uppi \) and also \(\sqrt {\uppi }\) with compass and straightedge. From the latter, it follows that squaring the circle, i.e., the construction of a square with the same area as a given circle with compass and straightedge, is impossible.