Applications of Galois Theory

Abstract

Now that we have developed Galois theory and have investigated a number of types of field extensions, we can put our knowledge to use to answer some of the most famous questions in mathematical history. In Section 15, we look at ruler and compass constructions and prove that with ruler and compass alone it is impossible to trisect an arbitrary angle, to duplicate the cube, to square the circle, and to construct most regular n-gons. These questions arose in the days of the ancient Greeks but were left unanswered for 2500 years. In order to prove that it is impossible to square the circle, we prove in Section 14 that π is transcendental over ℚ, and we prove at the same time that e is also transcendental over ℚ. In Section 16, we prove that there is no algebraic formula, involving only field operations and extraction of roots, to find the roots of an arbitrary nth degree polynomial if n ≥ 5. Before doing so, we investigate in detail polynomials of degree less than 5. By the mid-sixteenth century, formulas for finding the roots of quadratic, cubic, and quartic polynomials had been found. The success in finding the roots of arbitrary cubics and quartics within a few years of each other led people to believe that formulas for arbitrary degree polynomials would be found.