Vol. 6, 1920 
MATHEMATICS: H. S. VANDIVER 
267 
and if Ix denotes the real value of log x, then 
le{a), le{a'') 
D = 
le{a 
.le{c 
Hay"-') Ha-^ 
If €1 {a), €2(0;), 1(0:) are fundamental units in fi(a), that is, units 
which have the property that all units in the field can be expressed as 
products of powers of them multiplied by ^a^, then 
A = 
/€i(a) 
,/e^_i(a) 
To establish the above result Kummer essays to prove four direct theo- 
rems in the theory of the field ^{oc). These theorems are set forth later 
in this note and numbered I to IV. As will be shown, the proofs of theor- 
ems I and IV are incomplete as several unproved statements are made 
and the proofs of II and III are inaccurate, since use is made of a formula 
which is not true in general. 
In Crelle, 44, 1852 (134), Kummer gave the relation 
(X 
X — 2 
+ s 
Xi (a) 
(2) 
modulo X""^\ In this formula (p(a) is an algebraic integer in the field 
U{cx), and the symbol 
modulo X" \ represents that part of the logarithmic series 
(p{a) — <p{l) 
1 / <p{a) - <^(1) Y 1 ( i<pM - <P{1) \ 
3 
n + 1 
<^(1) 2 \ <^(1) 
none of whose terms are divisible by X 
for the norm of <^(a) and 
indicates that log (^(/) is to be differentiated 
4:uted for v in the result. Further, 
^(1) 
The symbol N(p{a) stands 
5 times and zero substi- 
X — 2 
Xi{a) = a + y-^^"a^ + y^^^^'a'^ + . . . . + 7" 
F. Mertens (Wiener Berichte, 126, 1917, Abt. Ila, pp. 1337-43) points 
out that the relation (2) is not true in all cases and shows that an error 
occurs in Kummer's argument on page 133 of the latter's paper. 
We shall now examine Kummer's 1857 memoir in detail and in the light 
of what has just been mentioned. 
