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Additional resources for Number Theory Noordwijkerhout 1983: Proceedings of the Journées Arithmétiques held at Noordwijkerhout, The Netherlands July 11–15, 1983

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Math. ( in press ). k Letter to D. Hayes, 9/15/1983. 32 6. : The Galois structure of S-units. to appear in the Sem. de Theorie de Nombres de Bordeaux. 7. Fr~hlich, A°: Artin root numbers and normal integral bases for q u a t e r n i o n fields. Inv. Math. 17(1972), 143-166. 8. : Galois module structure, in "Algebraic Number Fields," Proceedings of the Durham S y m p o s i u m 1975. Academic Press : London 1977, 133-191. 9. : Galois module structure of algebraic integers. S p r i n g e r - V e r l a g : Berlin, Heidelberg, New York, Tokyo (1983).

The proof is left to the reader. In the case A = Z s i n 2 17 P l ~ c h 2 ( r r x 1 4 . (s)-l[ estimates We begin here conjectures that heuristic the second be considered These on the zeroes ;~ , t h a t f o r A (s)/1 s to deduce real, arithmetic to a t l e a s t ! assumptions part of this paper. that all the statements all derive hence a n d it i s e a s y one would needmultiprecision to b e a b l e t o d e t e c t § 8. - T h e f u n d a m e n t a l it m u s t = ~ 1 e 2rr x 1 4 . 10 "38 . Hence 40 d e c i m a l s 1 Pl ~ ' ~ - + 1 4 .

2. - For k2i~ G / e l ~we i (el ~~k2(G/G1 ~ " this is just a r e s t a t e m e n t of p r o p o s i t i o n 3. 1, iv) . - Let K and C be f i n i t e A-modules. Z Wk(G ) ~ [ G I ~ < G : G I = ' K G up to A - i s o m o r p h i s m and G/GI~C ] P r o o f . - We c o n s i d e r o n l y k f i n i t e s i n c e the c a s e We s h a l l c o u n t the n u m b e r of p a i r s Ak/5~C rank , J / H ~-K . N o t e t h a t (H,5) H and Then for all k k =~ f o l l o w s by m a k i n g of A - m o d u l e s J Wk(K ) Wk(C ) = s u c h that are necessarily HcJcA k -, ~ .

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