4i6 
MATHEMATICS: H. S. VANDIVER 
Proc. N. a. S. 
ON THE CLASS NUMBER OF THE FIELD 12(^2.v/i.«) AND THE 
SECOND CASE OF FERMATS LAST THEOREM 
By H. S. Vandivkr 
Department of Mathematics, Cornell University 
Communicated by L. K. Dickson, March 31, 1920 
In the July 1919 number of the Bulletin of the American Mathematical 
Society, page 458, I gave an expression for the residue of the first factor 
of the class number of the field defined by e^^""^^, p being prime, with re- 
spect to the modulus in terms of Bernoulli numbers. In the present 
paper an analogous expression for the residue of the first factor of the class 
number of modulo p, will be obtained and the result used to 
show that certain results due to Bernstein^ on Fermat's last theorem do 
not have the generality stated by him. In view of the criticisms on 
Kummer's 1857 memoir on the last theorem which I have given elsewhere^ 
it is then pointed out that up to the present time no rigorous proof of the 
theorem for the exponents 59 and 67 has been given. 
Westlund^ reduced the first factor of the class number of ^{e^^^'/P^), if 
n>iy to the form 
/zi = ^ X ^ 5 = kk, (1) 
\'2f~\p-lY 'hnp^'-'ip-iy-l 
where k is the first factor of the class number of ^{e^'''^'^'), 8 = e^'''^'', 
=z ipimj = p^^~^{p—l), ni = p^, m^ = p"~^yri is the least positive 
residue of r\ modulo p", r being a primitive root of p^. The integer 5 
takes on all odd values < jjl except multiples of p, and the function (p{6) 
defined by Westlund may be put in the form 
<f{d) = To +nd + + 
We shall now reduce ki modulo p. To do this a modification of the method 
used by Kronecker in reducing the first factor of the class number of 
^(^2lV//>^J ^'ij employed. 4 We have 
(r-er'Me) = p^'gid) (2) 
where 
g{e) = go + gi^ + + q^-iO'-' 
and 
n - -^i + l 
P 
From an argument used by Westlund we also have 
where ju^ = (p{m^). Since r is a primitive root of then = — 1 + 
