In his monograph “Über die Klassenzahl abelscher Zahlkörper” [Berlin: Akademie Verlag (1952; Zbl 0046.26003), reprinted Berlin etc.: Springer Verlag (1985; Zbl 0668.12004)] H. Hasse obtained various results about class numbers and units of abelian number fields from an intensive study of the analytic class number formula. One of the central themes in “Class number parity” is the extension of some of Hasse’s results to relative abelian extensions using a mainly algebraic approach. The central tool replacing the analytic class number formula is an exact hexagon linking the cohomology of the class group and of the unit group of a cyclic extension of number fields: If \(E/F\) is a cyclic extension of number fields with Galois group \(G\), let \({\mathcal O}_ E\) denote the ring of integers in \(E\), \({\mathcal O}^*_ E\) the group of units and \(C(E)\) the class group. The authors establish the following exact hexagon of cohomology groups \[ \begin{alignedat}{7} &&&&&H^ 1(G,C(E))& \longrightarrow & H^ 1(G,{\mathcal O}^*_ E)\\ &&&\nearrow&&&&&&&&\searrow \\&R^ 0(E,F)&&&&&&&&&&&&R^ 1(E,F) \\ &&&\nwarrow&&&&&&&&\swarrow \\ &&&&&H^ 0(G,C(E))&\longleftarrow &H^ 0(G,{\mathcal O}^*_ E) \end{alignedat} \] where the groups \(R^ i(E,F)\), \(i=0,1,\) reflect ramification properties of the extension and — at least in the prime-cyclic case — are easily computable via class field theory.
Once the machinery is set up, which is done in Chapter 1, the authors look at various special situations:
In Chapter 2 relative quadratic extensions are studied with an emphasis on questions about the parity of class numbers and signs of units, a theme in the spirit of Weber’s classical result for the cyclotomic fields of 2-power roots of unity. It is in these applications that the hexagon reveals its power, since many classical results easily carry over to relative quadratic extensions.
In Chapter 3 the authors study those quadratic and biquadratic extensions over the rationals, which have odd class number, again a classical theme in the abelian case, but here a thorough treatment of non-abelian biquadratic extensions with odd class number is included.
It should be mentioned that the book contains many non-trivial examples, which illustrate the results and might turn out to be very useful in another context as well, e.g. in the study of similar questions in higher algebraic \(K\)-theory.