Number Theory in the 20th Century: Part 1

By “number theory” I mean algebraic number theory, and by Part 1, roughly the years 1900–1940. The treatment is very sketchy; I hope at some later date to cover the rest of the century and to fill in some of the gaps and correct any errors in this hasty account. At the end of the 19th century, Kurt Hensel introduced the idea of the completion kv of a field k with respect to a discrete valuation v and proved his famous lemma: If ov is the ring of integers in kv and f(t) is a monic polynomial with coefficients in ov, a factorization of f(t) into relatively prime factors modulo the maximal ideal of ov can be lifted to a factorization of f(t). At that time also, Hilbert, in his “Zahlbericht”, reinterpreted the quadratic reciprocity law as a product formula for his norm residue symbol. This product formula has been generalized to the higher K groups by Dustin Clausen. Let k be a number field, i.e., a field of finite degree over Q. Let m be an ideal in the ring of integers ok of k, let Im be the group of fractional ideals prime to m, and let Pm be the group of principal ideals (α) of elements α which are congruent to 1 mod m and positive at every real place. In the late 1800s Heinrich Weber had the idea that there should exist an abelian extension K of k such that the prime ideals of k which split completely in K are those in Pm, and the Galois group Gal(K/k) is isomorphic to Im/Pm. He called such an extension K or any subextension of it a class field over k. In 1907 Furtwängler showed the existence of such an extension in case m = 1, the so-called “Hilbert class field”. Near the end of the First World War, great progress was made by Teiji Takagi. He showed the existence of a class field for every m and that every abelian extension is a class field. A character of k defined mod m is a nonzero multiplicative map χ : Im → C∗ which is trivial on Pm. To such a χ, Richard Dedekind associated an “L-series”