Galois Theory, 5th Edition

Chapter 24Transcendental Numbers

DOI: 10.1201/9781003213949-24

Our discussion of the three geometric problems of antiquity—trisecting the angle, duplicating the cube, and squaring the circle—left one key fact unproved. To complete the proof of the impossibility of squaring the circle by a ruler-and-compass construction, crowning three thousand years of mathematical effort, we must prove that π is transcendental. (In this chapter the word ‘transcendental’ means ‘transcendental over $\mathbb{Q}$ ’.) The proof we give is analytic, which should not really be surprising since π is best defined analytically. The techniques involve symmetric polynomials, integration, differentiation, and some manipulation of inequalities, together with a healthy lack of respect ...