ON THE APPLICATIONS OF CYCLOTOMIC FIELDS IN INTRODUCTORY NUMBER THEORY 11 since there are precisely l quadratic residues modulo p quadratic character modulo p and since ( 0 p ) = 0. So f ( ζ ) = − ζ $^{l}$f ( ζ − $^{1}$). So writing f ( x ) = a$_{l}$x l + a$_{l}$ − $_{1}$x $^{l}$− 1 + ... + a$_{1}$x + a$_{0}$ , this yields a$_{l}$ζ l + a$_{l}$ − $_{1}$ζ $^{l}$− 1 + ... + a$_{1}$ζ + a$_{0}$ = − a$_{0}$ζ l − a$_{1}$ζ $^{l}$− 1 − ... − a$_{l}$ − $_{1}$ζ − a$_{l}$ , i.e. (3.4) l ∑ k =0 a$_{k}$ζ k = l ∑ k =0 ( − a$_{k}$ ) ζ $^{l}$−k Now it is trivial to see that a$_{l}$ = 2 by the above formula for f ( x ). Also, q$_{1}$ ( x ) = 2 ∏ $^{1}$≤ k<p ( k $_{p}$) =1 ( x − ζ $^{k}$). The constant term of q$_{1}$ is 2( − 1) l ∏ $^{1}$≤ k<p ( k $_{p}$) =1 ζ k = − 2 ∏ $^{1}$≤ $^{k}$≤l ζ k 2 = − 2 ζ l ( l +1)(2 l +1) 6 = − 2 ζ p $^{$_{p}$2}$− 1 24 . Now 3 | p 2 − 1 since p = 3 ( p ≡ 3 (4)), and p 2 ≡ 1 (8) since p is odd. So 3 · 8 = 24 | p 2 − 1. So The constant term of q$_{1}$ is − 2 · 1 = − 2. But q$_{1}$ ( x ) = f ( x ) + √ p ∗ g ( x ) where f ( x ) , g ( x ) ∈ Z [ x ]. So we must have a$_{0}$ = − 2. Therefore a$_{l}$ = − a$_{0}$ . So (3.4) now yields $^{l}$− 1 ∑ k =1 a$_{k}$ζ k = $^{l}$− 1 ∑ k =1 ( − a$_{k}$ ) ζ $^{l}$−k = $^{l}$− 1 ∑ k =1 ( − a$_{l}$ − $_{k}$) ζ k (after replacing k by l − k ), and { ζ, ..., ζ $^{l}$− $^{1}$} is a Z -linearly independent subset. So a$_{l}$ − k = − a$_{l}$ for 1 ≤ k ≤ l − 1, and so by the above, a$_{l}$ − k = − a$_{l}$ for all 0 ≤ k ≤ l . We can therefore rewrite f ( x ) as 2( x l − 1) + b$_{1}$x ( x $^{l}$− 2 − 1) + b$_{2}$x $^{2}$( x $^{l}$− 4 − 1) + ... + b $^{l}$− 1 2 x $^{l}$− 1 2 ( x − 1) = $^{l}$− 1 2 ∑ k =0 b$_{k}$x $^{k}$( x $^{l}$− 2 k − 1), b$_{k}$ ∈ Z for all 0 ≤ k ≤ $^{l}$− 1 2 (with b$_{0}$ = 2). ∏ $^{1}$≤ k<p ( k $_{p}$) =1 ( ζ − 1 − ζ $^{k}$). So f ( ζ ) f ( ζ − $^{1}$) = ∏ $^{1}$≤ k<p ( k $_{p}$) =1 ( ζ − ζ − $^{k}$) ( ζ 1 ζ $^{k}$) = ( − 1) l ∏ $^{1}$≤ k<p ( k $_{p}$) =1 ζ $^{1}$− k since ∑ $^{1}$≤ k<p ( k $_{p}$) =1 k = p p − 1 2 + 0 since the Legendre symbol is a = − ζ l ∏ $^{1}$≤ k<p ( k $_{p}$) =1 ζ − k = − ζ l