On a lower bound for the class number of an imaginary quadratic field
For an imaginary quadratic field K of discriminant D let h(K) be its class number and let P(x) be equal either to F(x) or to \(F(x)+x\), according to whether D is divisible by 4 or not, and \(F(x)=x^ 2+N(w)\) where \(w=D^{1/2}/2\), if \(D\equiv 0 (mod 4)\), and \(w=(1+D^{1/2})/2\) otherwise. The author considers \(p_ K\), the maximal number of prime factors of P(x) for \(x=0,1,2,..., | d(K)| /4-1\), and proves that h(K) is always at least equal to \(p_ K\) with equality occuring in the cases \(h(K)=1,2\). [The first case is equivalent to \textit{G. Rabinowitsch}'s theorem, J. Reine Angew. Math. 142, 153-164 (1913).] He notes also that for \(D=-84\) the numbers h(K) and \(p_ K\) do not coincide. (Reviewer's note: The name of Rabinowitsch occurs in this paper in two distinct spellings, both wrong).
scientific article; zbMATH DE number 3948414