On the parity of the class number of the field of q-th roots of unity

It is shown that the parity of the class number of the field Q(Cg) of the q-th roots of unity over the rationals is odd whenever q and p = (q -l)/2 are primes and 2 is inert in the real subfield of p-th roots of unity over the rationals.As a consequence, the genus coincides with the spinor genus of the ring of integers in Q(Cg) viewed as a lattice over the ring of integers in the real subfield.Throughout this article, £ n denotes a primitive n-th root of unity, kn = Q(Cn) is the cyclotomic field of n-th roots of unity over the rationals Q; fc+ denotes the real subfield of k n ; 0+ and 0+ are the rings of algebraic integers in k n and k+ ; C n , C+ are the class groups of the two rings; and their orders /i n , h" are the class numbers of the two fields.It is known that /i+ divides h n with quotient h~, the relative class number of k n over k+.Moreover, h n is odd if and only if h~ is odd [8, Satz 45].We prove in this note the following THEOREM.If q and p -(q -l)/2 are prime integers and 2 is inert in the real subfield of the cyclotomic field of p-th roots of unity over the rationals then the class number of the cyclotomic field of q-th roots of unity is odd.The study of the parity of the class number of number fields is motivated by the research of several authors.H. Hasse credits E. Kummer with the initial investigations on the parity of h q ,q a prime, based on a series of Kummer's papers between 1847 and 1870 (See [8] for a list of Kummer's articles and Satz 45 for extensions and refinements of Kummer's results to imaginary cyclic extensions of Q).Kummer's work can be viewed as an analogue of Gauss' work on the 2 primary component of the class groups of binary quadratic forms for the binary quadratic lattice O q over 0+ (see Proposition below).In