Cyclotomic Fields of Class Number One

In this chapter we determine those m for which $$ \mathbb{Q}({\zeta _m}) $$ has class number one. In Chapter 4, the Brauer–Siegel theorem was used to show that there are only finitely many such fields, but the result was noneffective: there was no computable bound on m. So we need other techniques. Since h_n divides h_m if n divides m, it is reasonable to start with m prime. In 1964 Siegel showed that h_p = 1 implies p ≤ C, where C is a computable constant, but the constant was presumably too large to make computations feasible. In 1971, Montgomery and Uchida independently obtained much better values of C, from which it followed that $$ {h_p} = 1 \Leftrightarrow p \leqslant 19 $$ Masley was then able to use this information, plus a table of h_m- for ø(m) ≤ 256, to explicitly determine all m with h_m=1.