On the Constructability of Regular n-Gons

Numbers like \(\sqrt {2}\) and \(-\sqrt {2}\) are algebraically indistinguishable (over the rational numbers), as both numbers have the same minimal polynomial, namely \(X^2 - 2\) . We call such numbers Galois conjugates of each other. We show that two Galois conjugate numbers not only have the same minimal polynomial, but that even every polynomial equation that is satisfied by one number is also satisfied by its Galois conjugates and vice versa. In general, we call polynomial relationships between several algebraic numbers algebraic relations. Of particular interest are the algebraic relations between the roots of a polynomial. If we number the roots, we call a permutation of the roots a (Galois) symmetry if all algebraic relations between these roots are preserved under this permutation. All symmetries together form the Galois group of the roots. From the definition of the Galois group, it is not immediately clear how we can effectively calculate it. This is done using Galois resolvents, and we provide a complete procedure in this chapter. By being able to assign a group to each (the roots of each) polynomial, we can in turn draw conclusions about the polynomial and its roots from the group structure. Therefore, in this chapter, we look at some very general statements about groups, such as the class equation. We apply these theorems to the Galois groups of the so-called cyclotomic polynomials. As an application, we provide a complete classification of the regular n-gons that can be constructed with compass and straightedge.