The Development of the Principal Genus Theorem

Genus theory belongs to algebraic number theory and, in very broad terms, deals with the part of the ideal class group of a number field that is ‘easy to compute’. Historically, the importance of genus theory stems from the fact that it was the essential algebraic ingredient in the derivation of the classical reciprocity laws – from Gauß’s second proof over Kummer’s contributions up to Takagi’s ‘general’ reciprocity law for p-th power residues. The central theorem in genus theory is the principal genus theorem, which is hard to describe in just one sentence – readers not familiar with genus theory might want to glance into Section 2 before reading on. In modern terms, the principal genus theorem for abelian extensions k/Q describes the splitting of prime ideals of k in the genus field kgen of k, which by definition is the maximal unramified extension of k that is abelian over Q. In this note we outline the development of the principal genus theorem from its conception in the context of binary quadratic forms by Gauß (with hindsight, traces of genus theory can be found in the work of Euler on quadratic forms and idoneal numbers) to its modern formulation within the framework of class field theory. It is somewhat remarkable that, although the theorem itself is classical, the name ‘principal ideal theorem’ (‘Hauptgeschlechtssatz’ in German) was not used in the 19th century, and it seems that it was coined by Hasse in his Bericht [29] and adopted immediately by the abstract algebra group around Noether. It is even more remarkable that Gauß doesn’t bother to formulate the principal genus theorem except in passing: after observing in [26, §247] that duplicated classes (classes of forms composed with themselves) lie in the principal genus, the converse (namely the principal genus theorem) is stated for the first time in §261: si itaque omnes classes generis principalis ex duplicatione alicuius classis provenire possunt (quod revera semper locum habere in sequentibus demonstrabitur), . . . 1 The actual statement of the principal genus theorem is somewhat hidden in [26, §286], where Gauß formulates the following Problem. Given a binary form F = (A,B,C) of determinant D belonging to a principal genus: to find a binary form f from whose duplication we get the form F . It strikes us as odd that Gauß didn’t formulate this central result properly; yet he knew exactly what he was doing [26, §287]: