FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. 
59 
[721 
721] FORMULA INVOLVING THE SEVENTH ROOTS OF UNITY. 59 
where \Jr (a) is a rational and integral function of a with integral coefficients; or, what 
is the same thing, if a and /3 be any two roots of the above-mentioned equation, then 
F(a)F(P) = + (a, /3)F(«/3), 
asmuch as the 
ain the proper 
thus obtain 
where л}г (a, /3) is a rational and integral function of a, /3 with integral coefficients. 
As regards the proof of this, it may be remarked that, writing x 3 for x, F{a), F(/3), 
and F(a/3) become respectively cr l F{a), /3 _1 F(/3), (a/3)~ l F(a/3); hence, F(a) F (/3) -r- F (ot/3) 
remains unaltered, and it thus appears that the function in question is expressible 
rationally in terms of the adjoint quantities a and /3. With this explanation the 
■2a,)}, 
following extract will be easily intelligible: 
2a,)}, 
“ The true form (never yet given) of the roots of the equation x p — 1 = 0 is as 
follows: The roots, as is known, can easily be expressed by mere addition of the 
functions F(a). If X is a factor of p— 1 and a A = 1, then it is further known that 
3a,)}, 
{F(a)} A is a mere function of a. But it is only necessary to know those values of 
F(a) for which X is the power of a prime number. For suppose XX'X"... is a factor 
3a,)}, 
of p — 1; further let X, X', X",... be powers of different prime numbers, and a, a, a",... 
prime Xth, X'th, X"th, ... roots of unity, then 
®)}> 
F(aaa ...)- ^ ^ a „ } 
®)}; 
where -yjr (a, a', a",...) denotes a rational and integral function of a, cl', a",... with 
integral coefficients. Hence, considering always the (p — l)th roots of unity as given, 
motion of the 
its which are 
there are contained in the expression for x only radicals, the exponents of which are 
powers of prime numbers, and products of such radicals. But if X is a power of a 
prime number, — p n , suppose, the corresponding function F(a) can be found as follows: 
Assume 
&> 2 + a, + 1 = 0, 
i for 3r, and 
,btain for 9r 2 
ssion for 3r 2 ; 
Qg, we should 
equations we 
F (a) F (cd) = \[г { (a) F (а г+1 ): 
then 
F{a) =^/{л/г а (а) л\г, (a) a) F«)}, 
F (a*) = у/{^i (a'*) ^ (or) ... (a?) F(a'" 2 )}, 
and so on, up to 
p-1 
F(a' 4 ”' 1 ) = (/{A-i (a^-j л/г 2 (a^j ... yfr^ (a^ _1 ) (-) M p},” 
paper “Ueber 
r Monatsber., 
¡—274]. The 
ccP-1 _ 
m = 0, 
x—1 
so that the formulae contain ultimately /¿th roots only. It is remarked in a foot 
note that, when n = 1, the p—1 functions can always be reduced to one-sixth part in 
number, and that by an induction continued as far as ¿4=31, Jacobi had found that 
all the functions \Jr could be expressed by means of the values of a single one of 
these functions. 
“ The /4 — 1 functions determine, not only the values of all the magnitudes under 
the radical signs, but also the mutual dependence of the radicals themselves. For 
replacing a by the different powers of a, one can by means of the values so obtained 
for these functions rationally express all the ¡x n — 1 functions F(a 1 ) by means of the 
powers of F(a); since all the F n — 1 magnitudes (F(a)} i ч- F (a. 1 ) are each of them 
8—2