# De nombres remarquables by Francois Le Lionnais

By Francois Le Lionnais

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Satisfies of the o . 4) f . 5) ~' eu(~) = ~' u u UW (I- ~ d d ~q -1 (T~,c) c now taking the change the a . Let 6 aN product order of N >> over product 0 . Then )) aCa = fl-q the RS and take {c} the . ~t c of a/fa partial (~,af-c) ~ af a . We product ' may inter- over there lemma. ,gn_1 n eu(W) e ~g, sg Let eu(~) where g put isomorphism ~gg' we replace . 6) , and changing and c o l l e c t i n g in the second anything. 7) sgn(f) -j s **/q 1-g-r+N+d = q sgn(a) , a - [' deg b + dj]/6 and , where a through aN , . **

1) suggests to look for such ~' B . 8 4 Proposition that 4' s is u n i q u e l y smallest over which field be defined. Now s 6 C Corollary. equals I, 4' with w-th H(s q-l) 4' The has with be d e s c r i b e d § 4]. 11 . sw 6 H such roots of u n i t y . 4) Corollary. 3]. T h e r e e x i s t s -I is s g n - n o r m a l i z e d . 4' field over see b e l o w . 9 does ~ in B . Each 4' H(s q-l) . 12 field with = and : H 4 . it h a s this sgn-normalized H(D(4,n)) AUtA(D(4,n)) with coefficients over H , and with its G a l o i s group.

See numbers the the of K(a) . sub- are . 8] analogy fields x of [I class as the x K* • K* • K* first In f a c t , If with a lattice (q-1)-st root A = ~ " A , the of u n i t y . 5 Remark. corresponds pressions In t h e to the for ~ 1-~(q-1)q i~I I-~ (qi+1-1) above mentioned number ~ are ~q-1 = ~ and known, for (Tq_T) I' analogy ~ to of K with 2~i . Several residue field ~ , t° further ex- instance al-q a6A or the product 2. 10) over [39, again Fq A . § 4] be arbitrary. , and resp. has E (I)~ The a uniquely is t h e group determined of u n i t s k at lifting resp.