The Class Number of the Cyclotomic Field.

1. Let g denote any odd prime and h = h(g) the class number of the cyclotomic field R(r), where r is the primitive gth root of unity, R the rational numbers. It is known that we can write: h = h1h2, where hi and h2 (both integers) are the so-called first and second factors of the class number; in fact h2 is the class number of the real field of degree 2 under R(r), namely the field R(D + D-). Kummer conjectured (J. Math., 16, 473 (1851)) that g(g+3)/4 hl t-2(9-3)/2 (g-1)/2 = G. (1)