The Fundamental Theorem of Algebra

The polynomial equation \(X^2 + 1 = 0\) has no solution in the ring of real numbers, but it does in the ring of the so-called complex numbers. In this chapter, we introduce the complex numbers \(\mathbf {C}\) as the smallest possible extension of the ring of real numbers in which the aforementioned polynomial equation has a solution. Just as we can identify the entirety of real numbers with the number line, we can identify the entirety of complex numbers with the number plane. Complex numbers are thus points in a plane. If we mark the numbers 0 and 1 in the plane, we call a complex number constructible if the corresponding point can be constructed with a compass and straightedge from the two marked points. Questions about the feasibility of certain construction problems—such as squaring the circle—can thus be reduced to algebraic statements about complex numbers. It is an astonishing fact that in \(\mathbf {C}\) not only \(X^2 + 1 = 0\) has a solution, but automatically every polynomial equation \(X^n + a_{n - 1} X^{n - 1} + \dotsb + a_1 X + a_0 = 0\) with rational coefficients \(a_0\) , …, \(a_n\) , \(n \ge 1\) . This is the statement of the so-called fundamental theorem of algebra. At the end of this chapter, we provide a constructive proof for this theorem, which goes back to Martin Kneser. A complex number that is a solution of such a polynomial equation with rational coefficients is called algebraic. In this chapter, we show, among other things, that constructible numbers are always algebraic, which will be a key for the proof of the impossibility of squaring the circle.