By Leonard D. Baumert
Publication by means of Baumert, Leonard D.
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Extra resources for Cyclic Difference Sets
Proofs may be found, for example, in Mann's book (1955). 18. degree ~(d). The numbers ~(r) K(~d) The conjugates of i~ ~d' rationals. any The field If ~(d)-! °'''~d (r,s) = 1 is a normal extension of the rationals of ~d are the ~(d) form an integral basis for the field consecutive powers of K(~rs ) ~r i ~d numbers where K(~d) is of degree ~(r) (i,d) = 1. 19 . 46) (P) = P! 47) p Pi are conjugates. modulo g is d. 48) can be determined explicitly. irreducible equation satisfied by Further, The field automorphism determined by the fixes each of these prime ideals (P) = (P1 "'" pg)~(pa) where is given by when is the order of ~d ~ ~ in [that (p) = (i - ~d )~(d) where the distinct prime ideals where p The prime ideal decomposition of the rational prime modulo For, if w.
59 (7,v) = i Note that in particular this eliminates the condition result. For if residue of 7 n = 2p a (a odd), the assumption that implies that contradiction that a t3 ~ p3jp ~ -i (mod 7). 7 provides the is necessarily even. Mann and Zaremba (1969) investigate the situation when is not a quadratic residue of 7. 7 divides v and However they do not resolve it completely. particular, they find no case where t t In is not a multiplier. On occasion it is possible to establish the existence of a w-multiplier for some divisor multiplier.
1 it follows odd. 19) , it follows necessarily f. In this later case and all integers is w-multiplier, From this and the prime n 2 > i. for some integer q is another prime divisor of t r. of [8((w)]2 with q. w s > 1 is a quadratic if Unless modulo v, k, ~ - difference 4 for all multipliers of of a non-trivial is odd w = qS Proof. is: this yields n = m q. with and q = i n = 4q n -= v = 0 2 d > 1 modulo q q Thus (d,q s) = l, which contradicts with as prime. As that q2 the 31 divides n; thus with is not assumed that As before, in nO = nl/q the desired form 2 3 n = noq emerges, where it (no, q ) = 1.