Bernoulli Numbers

Abstract

In this chapter we will introduce an important sequence of rational numbers discovered by Jacob Bernoulli (1654–1705) and discussed by him in a posthumous work Ars Conjectandi (1713). These numbers, now called Bernoulli numbers, appear in many different areas of mathematics. In the first section we give their definition and discuss their connection with three different classical problems. In the next section we discuss various arithmetical properties of Bernoulli numbers including the Claussen—von Staudt theorem and the Kummer congruences. The first of these results determines the denominators of the Bernoulli numbers, and the second gives information about their numerators. In the last section we prove a theorem due to J. Herbrand which relates Bernoulli numbers to the structure of the ideal class group of ℚ(ζ_p). The material in this section is somewhat sophisticated but we have included it anyway because it provides a beautiful and important application of the Stickelberger relation which was proven in the last chapter.