NOVEMBER 12, 1896] 
NATURE 27 
is then called the Galoisian group of the corresponding 
equation. 
The problem of the algebraical solution of an equation 
ultimately depends upon the nature of its Galoisian 
group, or, which comes to the same thing, of its Galoisian 
resolvent. The degree of this resolvent is equal to p, 
the degree of the Galoisian group ; if this is equal to 7 ! (7 
being the degree of the proposed equation), the equation 
is said to have no Affect. So long as we confine our- 
selves to the corpus Q(a@, @,..-.a@n-), the general 
equation 
Gp" + a2". |. A dn =O 
has no Affect. But, by the adjunction of an appropriate 
algebraical quantity, the Galoisian resolvent may become 
reducible, and then any one of its irreducible factors is 
a Galoisian resolvent in the new domain of rationality. 
The process may admit of repetition, and we may say 
that the problem ultimately consists in finding alge- 
braical quantities of the simplest possible kind, so that 
by their successive adjunction we may obtain a series of 
resolvents of lower and lower degree. No such reduc- 
tion can be effected by the adjunction of irrationalities 
which are notcontained inthe normal corpus Q(2), 2, . . . 21) 
which is derived from the original equation. After the 
proof of this very important result (p. 516), the chapter 
concludes with a further discussion of imprimitive groups, 
with special reference to the Galoisian resolvent. 
Chapter xv., on cyclical equations, contains applications 
of the foregoing theory, and should be read concurrently 
with chapter xiv. by those to whom Galois’s theory is 
new. First of all, the general cubic and biquadratic are 
solved by a direct application of group-theory ; and we 
are then introduced to the theory of Abelian and cyclic 
equations. An Abelian equation is a normal equation 
whose Galoisian group is commutative. It is not neces- 
sarily irreducible; on the other hand, an irreducible 
equation with a commutative group is necessarily Abelian 
(p. 535). 
A cyclical equation is one whose group consists of a 
single cyclical substitution and its powers; in other 
words, when its roots may be arranged in such an order 
that all cyclical functions of them are rational. The 
solution of any Abelian equation may always be reduced 
to that of a system of cyclic equations (§ 163). The 
chapter concludes with the solution of cyclic equations 
by means of Lagrange’s resolvent. 
Chapter xvi., on cyclotomy, gives the theory of Gauss’s 
periods, of the auxiliary functions which Jacobi denotes 
by y,(a), of the solution of cyclotomic equations by their 
means, and of Gauss’s sums. It concludes with appli- 
cations to complex numbers of the forms x + yz, x + yp, 
p being a cube root of unity. It may be observed that 
the properties of the numbers ¥,(a) are very fully treated 
in Jacobi’s lectures on the theory of numbers. These 
are not included in his collected works ; but MS. copies 
of Rosenhain’s redaction of them may be picked up 
occasionally. 
Chapter xvii. contains a series of remarkable pro- 
positions, which are proved with comparative ease by 
means of the foregoing theory. First of all it is shown 
that if P, the group of an equation, is reduced by the 
adjunction of the roots of an Abelian equation, P has a 
NO. I41I, VOL. 55] 
normal divisor Q, the index of which is a prime; and, 
conversely, if P has a normal divisor of this kind, P may 
be reduced in the manner stated. (By a normal divisor 
of P is meant a self-conjugate sub-group, or, as 
calls it, an “ausgezeichnete Untergruppe.”) 
We then pass on to the theory of metacyclic equations. 
A metacyclic equation is defined as one whose complete 
solution may be made to depend on that of a series of 
cyclic equations. It is shown (p. 598) that the necessary 
and sufficient condition to be satisfied by a metacyclic 
equation is that there should exist a series of groups 
RP MIPA at 
(of which the first is the Galoisian group of the equation) 
such that each group is a normal divisor, with prime 
index, of the group immediately before it in the series. 
It is subsequently proved (§ 180) that the group of a 
metacyclic equation of prime degree is linear ; that is to 
say, if its roots are 
eal Coteienet Coiene! =) Dp 
Klein 
the group consists of the A(#—1) permutations of the 
suffixes defined by 
’=at + 6(mod f) , 
where a may have any of the values I, 2,... 
any of the values 0, 1, 2... (f—1). 
Conversely every irreducible equation of prime degree 
whose group is linear, is metacyclic (p. 615). 
A function of the roots 2; which is unaltered by the 
permutations of the linear group, and by these only, is 
called a metacyclic function. Every metacylic function 
is a root of a rational equation F(y)=o of degree (7—2)! 
and the function y may be so chosen that this equation 
has no multiple roots. Assuming that y has been so 
chosen, the necessary and sufficient condition that a given 
equation /(x)=o may be soluble by radicals is that the 
resolvent F(y)=o has a rational root (p. 620). This very 
beautiful proposition is the complete answer to the ques- 
tion first definitely put by Abel, namely : What algebraical 
equations are soluble by radicals ? 
At the end of this chapter will. be found a very 
interesting application to metacyclic quintics. Among 
other things it is shown that if A and p are any two 
rational quantities, and 
ss, SIA 
*= (a — 1)i(A2 + 6A +25), 
the quintic 
(p—1), and 4 
5°wA 
(A —1)*(A + 6A + 25) 
x +axr+B=0 
may be solved by radicals. 
One problem still remains: the actual construction of 
all possible metacyclic equations. This is considered in 
chapter xviii, the last in vol. I. The complete solution 
was announced by Kronecker, in his usual oracular way, in 
the Berlin Wovatsberichte for 1853 and 1856. Prof. Weber 
here supplies us with a demonstration, which reproduces 
in an improved and simplified form his own Marburg 
memoir of 1892. It is based upon the properties of 
Lagrange’s resolvent ; and although it is impossible to 
analyse it in detail, the final result may be stated. 
“ Every root € of a metacyclic equation of prime degree 
n may be expressed in the form 
s=n-2 
g=A+>d 
s=0 
5 > . nf), r ae al 
where A is a rational quantity, t» = N/4s, K a rational 
rn—2 rn-3 ry 
- _n-2 
Ky; Te+1 + Ts+u-2 
