268 
MATHEMATICS: H. S. VANDIVER Proc. N. A. S. 
Under assumption 1, given above, namely, that hi is isdivible by 
X but not by X^, Kummer proceeds to show that only one of the numbers 
Bi, i = 1, 2, , iJL — 1, is divisible by X. In this connection he states 
(p. 42, 5th line from bottom) "... .denn der erste Faktor der Klassen- 
anzahl muss, wie aus meiner Untersuchung der Teilbarkeit der Klassen- 
anzahl durch X {LiouvilWs Journal, 16, 1851 (473)) unmittelbar folgt, 
den Faktor X mindestens so viel mal enthalten, als wie viele der (X — 3)/2 
Bernoullischen Zahlen durch X theilbar sind." On examining the paper 
referred to by Kummer we find that he reduces hi modulo X and obtains 
a residue which may be expressed in the form 
mliBi 
i = 1 
where m is an integer prime to X. At no place in this work however, does 
he refer to any reduction of hi, modulo X", n>\. Hence, he is not war- 
ranted in making, without additional argument, the statement given 
above. 
Having concluded that only one of the first )U — 1 Bernoullian numbers 
is divisible by X, Kummer calls this one and derives (p. 44), under as- 
sumption 1, the following: 
Theorem I. If B^ = 0 (mod X), and = 0 (mod X) then EXa) is the 
X" power of a unit in 12 (a). 
On page 45 an attempt is made to prove, under assumption 1, the 
Theorem II. If B^= 0 (mod X) but B^x = 0 (mod X^) then a unit in 9.{a) 
which is congruent modulo X^ to a rational integer is the power of a unit 
in Q{a). 
In the argument covering this (p. 46), there is used the incorrect formula 
(2) of this paper. Also, on page 50, under assumption I, is given the 
Theorem III. If F (a) is an integer in the field ^l{a -\- a~'^) and if 
do^'lFie") 
—^^ — ■ = 0 (mod X) 
av" 
then a unit E{a) in 12 (cc) can be found such that 
E{a)F{a) ^ a (mod X) 
where a is a rational integer. 
In proving the above, the incorrect relation (2) is again used. 
In paragraphs 4, 5 and 6, pp. 50-61, Kummer goes through a course of 
reasoning which leads him to 
Theorem IV. // P is an ideal such that is the principal ideal (F{a)), • 
and Bj, = 0 (mod. X) then P is or is not a principal ideal according as 
do^'-^nFie') + 
— = or 0 + (mod X). 
In this connection note the formula {A) which he gives on page 53. 
The number ^^(q;) which appears here is defined as the product of certain 
