787 
A YP 
groups according as the LeGENDRE symbol (=) has the value +1 
m 
or —1. This implies an arrangement of the roots of /, (z). We have 
2mie 
— on (iH 
the rootsu—e ™ , where{ — |= 1 and an equal number of roots 
m 
2zic! ; 
wen ‚ where 6 , ed 
m 
Now by proposition VIJL we have 
Suk + Duk =p(m)u(D) g (LY), 
u u 
but at the same time we infer from Gauss’s theorem 
1 
= any ie — (ml)? 
Sut —- Su? — Tf — [44 ym. 
u u' m 
Hence the sums 2 uk and 2 wk may be calculated separately 
u u' 
and if we introduce the conjugate (real or complex) irrationalities 
1 \ 
ee ae } 
\ F 
y= 4 gum) + 74 Vim , 
\ (nl)? / 
aj’ = 4 pu(m) — i4 mi, 
it will be found that there exists a polynomial f”(«, 4) == (ru), 
u 
linear in 9 with real integer coefticients, having the roots « and 
also a quite similar conjugate polynomial f(z, 1) = M(& — v)), 
u 
having the roots 1. 
As obviously 
Fan (a) <= re (4,1) x Jan (wv, 1) 
it appears that by adjoining the irrationality 1 to the set of real 
integers tbe polynomial F’, (x) has become decomposable. 
BB ey te, 
The values of Zuk and Su'*t for :—1, 2,3,..., EN being calcu- 
u u = 
lated, it would be possible to find the coefficients of either of the 
polynomials f, (#, %) and /,, (#, 1), but I will only apply Gauss’s 
theorem to deduce a tolerably regular expression for their product 
Geh 
If. we substitute for « suecessively the roots w and w' in the 
identity 
A=M 
an Fe) —_ > Ayth*r, 
h=0 
