On the History of the Artin Reciprocity Law in Abelian Extensions of Algebraic Number Fields: How Artin was Led to his Reciprocity Law

Abstract

Artin’s Reciprocity Law (1923/26) for Abelian extensions of algebraic number fields is the central theorem of Class Field Theory which, by the Theory of Takagi (1920) [64], is the Theory of Abelian Extensions of Algebraic Number Fields. Abelian extensions of algebraic number fields have been studied extensively in the second half of the 19-th century,in particular by Kronecker(1853,1856,1877,1882), Weber (1886/87, 1897/98) and Hilbert (1896, 1897, 1898) who laid the foundations and discovered many fundamental properties of the Class Fields. These discoveries were made possible by the thorough study of a particular kind of Abelian extensions, namely the study of cyclotomic fields initiated by Gauss (1801) [24, Sect. 7] and carried further by Kummer (1847-1874). The term Abelian in connection with algebraic extensions (or, at that time, algebraic equations) was coined by Kronecker in 1853, first related to algebraic equations as ‘Abelian equation.’ 1 By this Kronecker referred to polynomials with cyclic Galois group. However, Kronecker was already then aware of the more general polynomials with Abelian (in the modern sense) Galois group, since he is referring on page 6 (in [45], Werke, Bd. 4) to the fundamental treatise of Abel of 1829 which appeared in Volume 4 of Crelle’s Journal [1], just two months before Abel’s death, and where Abel states explicitly the condition for commutativity θθ 1x =θ1θx in his assertion that a polynomial with Abelian Galois group is solvable (with radicals). Abel says there on page 479 of [1]: “En general j’ai demontre le theoreme suivant: ‘Si les racines d’une equation d’un degre quelconque sont liees entre elles de telle sorte, que toutes ces racines puissent etre exprimees rationnellement au moyen de l’une d’elles, que nous d’signerons par x; si de plus, en designant par θx, θ1 x deux autres racines quelconques, on a θ1 x = θθ 1 6x, 1’ equation dont il s’agit sera toujours resoluble algebriquement.’ ” (In general I have demonstrated the following theorem: If the roots of an equation of any degree are related to each other in such a way that all these roots can be expressed rationally by means of one of them, which will be denoted by x; if, in addition, we have θθ1x = θ1θx, where we denote by θx, θ1 x any two other roots, then the equation in question is algebraically solvable [that is, solvable by radicals].) Later in 1877 [48] Kronecker used the term ‘Abelian equation’in the larger sense to mean a polynomial with Abelian Galois group in our modern sense. He says there on page 66 (in [48], Werke, Bd. 4) that he now calls ‘Abelian equation’ an equation having the property that all its roots x are rational functions of any one of them and if θ 1 and θ2 are two of these functions then θ1θ2x = θ2θ1x. He then calls ‘simple Abelian equation’ the equation he treated in 1853, namely the equation with cyclic Galois group of prime order. That he had already used the term ‘Abelian equation’ in the paper of 1853 in the special case of cyclic equations is justified by Kronecker by the fact that the ‘Abelian equation’ can be reduced to the ‘simple Abelian equation,’ or as Kronecker says on page 69 (in [48], Werke, Bd. 4), that every root of any Abelian equation is a rational function of roots of simple Abelian equations.2 This justification is repeated in his paper of 1882 on the composition of Abelian equations.3 In the paper of 1853 Kronecker also states the famous theorem that any root of a (simple) Abelian equation [that is a cyclic equation] with integral rational coefficients can be represented by roots of unity, that is, is contained in a cyclotomic field over the rational number field.4 The same theorem is stated by Kronecker in his paper of 1877 in the case of the general Abelian equation with integral rational coefficients.5 It was later proved by Weber (1886) [67] and more simply by Hilbert (1896) [41]. In both papers Kronecker also stated his Jugendtraum for an analogous theorem for Abelian equations with coefficients in a quadratic imaginary number field. This theorem was proved partially by Weber (1908) [69] and Fueter (1914) [23] and completely by Takagi (1920) [64]. Both theorems have plaid a crucial role in the history of class field theory.6