The Cassels heights of cyclotomic integers

Abstract

AbstractWe study the set $${\mathscr {C}}$$ C of mean square values of the moduli of the conjugates of all nonzero cyclotomic integers $$\beta $$ β . For its kth derived set $${\mathscr {C}}^{(k)}$$ C ( k ) , we show that $${\mathscr {C}}^{(k)}=(k+1){\mathscr {C}}\,\, (k\ge 0)$$ C ( k ) = ( k + 1 ) C ( k ≥ 0 ) , so that also $${\mathscr {C}}^{(k)}+{\mathscr {C}}^{(\ell )}={\mathscr {C}}^{(k+\ell +1)}\,\,(k,\ell \ge 0)$$ C ( k ) + C ( ℓ ) = C ( k + ℓ + 1 ) ( k , ℓ ≥ 0 ) . Furthermore, we describe precisely the restricted set $${\mathscr {C}}_p$$ C p where the $$\beta $$ β are confined to the ring $${\mathbb {Z}}[\omega _p]$$ Z [ ω p ] , where p is an odd prime and $$\omega _p$$ ω p is a primitive pth root of unity. In order to do this, we prove that both of the quadratic polynomials $$a^2+ab+b^2+c^2+a+b+c$$ a 2 + a b + b 2 + c 2 + a + b + c and $$a^2+b^2+c^2+ab+bc+ca+a+b+c$$ a 2 + b 2 + c 2 + a b + b c + c a + a + b + c are universal.