ursh Journl of Anlyss n Nuber heory, 4, Vol., No., -8 Avlble onlne t htt://ubs.sceub.co/tnt///4 Scence n Eucton Publshng OI:.69/tnt---4 Nubers Relte to Bernoull-Goss Nubers Mohe Oul ouh Benough * érteent e Mthétue-Infortue, Unversté es Scences, e echnologe et e Méecne, Nouchott, Murtne *Corresonng uthor: ouh@unv-nc.r Receve Jnury, 4; Revse Februry, 4; Accete Februry, 4 Abstrct In ths er, we generlze Goss result ere n ([5], ge 5, lne 9, for, n gve chrcterzton of soe nubers of Bernoull-Goss [5] by ntroucng the secl nubers M(. Keywors: Bernoull-Goss, Crltz Moule, congruence, rreucble olynols. Cte hs Artcle: Mohe Oul ouh Benough, Nubers Relte to Bernoull-Goss Nubers. ursh Journl of Anlyss n Nuber heory, vol., no. (4: -8. o:.69/tnt---4.. Introucton et be fnte fel of s the chrcterstc of, n >. et B( n [ ] n eleents,,, onc enotes the n-th Bernoull-Goss nuber [5] whch s secl vlue of the zet functon of Goss n s n [ ]. n In the followng we gve chrcterzton of onc rreucble olynols vng B ( by ntroucng the nubers M(, for,,.. efntons n Nottons In ths secton, we ntrouce soe efntons n notton tht wll be use throughout the er. s fnte fel of eleents, s ower of re, ; A [ ], (, ; A onc n A ; { } et P A, we sy tht s re f P A n s rreucble; vp (. s the P c vluton where s re ;, [ ] ; n [ ] [ ],,. [] ;,,.[ ] [] n. [ ]. Crltz Moule et ρ be the Crltz oule whch s orhs of -lgebrs fro [ ] nto the - enoorhss of X X X the tve grou gven by. ρ, for n n n n A, n n ρ( X ρ ( X X, A, n for α, ρα ( X α. X... e ([5], Prooston.. et An,, then Where [ ]... e for et A {} of egree n, then [ ]. eg[ ] ( n f n. f n he roof s very esy n cn be one wth the followng Hnts:. By nucton on. hs s obvous.... e et P be re of egree n let n, then n P vp n, f n
4 ursh Journl of Anlyss n Nuber heory he roof cn be one by nucton on. 4. A rerble Congruence 4... efnton et,, we set S A,eg E ( X X n for,.. E X ( X A,eg < We hve : ([5], heore..5, E l, n usng Crltz's theore l E ( X ( X l ( l l l Now, we resent our frst theore whch generlzes result of Goss ere n [5], ge 5, lne 9 for. 4... heore et, then We hve (, S ( ( l E X E X ( X A,eg On the other hn, we hve : ( ( E X X So the logrthc ervtve of E ( X hus s: ( E ( X X A,eg herefore: n n n.. ( X X X n A X,eg n n ( X n A,eg n n n ( X S( ( n On the other hn, we hve : ( n Snce. E X E ( X E ( X. (. (. ( E X ( X E (. X ( We euce tht (. E ( X (. X o( X ( By entfcton, we obtn: ( S ( ( ( ( S ( hs terntes the roof. herefore : S ( 4... efnton (. X o( X ( ( ( ( ( We efne the -th Bernoull-Goss nubers s follows: B n, o ( B S A f B S A, f o,, 4..4. heore ([], heore et be re of egree, n c, then. c B c o( P c We hve
ursh Journl of Anlyss n Nuber heory 5 herefore c c o. ( ( B c S c For, n, {,, } n enote l. n, we Accorng to Shets ([9], we hve f l( < (, therefore S (, thus for,s c. Hence, t follows tht: ( ( B c S c ( S c o(p J So ccorng to heore 4.., we hve : ( ( B c S c hs terntes the roof. 4..5. e S c P c. ( o( P c ( ( o et P be reer of egree, then P ( o ( P, for P hs cn be shown by cobnton of n nucton on, n le. Now, we resent the followng rerble congruence: 4..6. heore([],heore et P be reer of egree, then ρ ( o ( P B( o P We hve ρ P P P P P ( P o ( P P B o ( P Snce for, c, we hve by heore 4..4 (. ( B B, n o B ( ( ( P o P B o( P 5. he Nubers M( We note tht : 5.. efnton For, we set ( (. M (, M ( M A n eg M Accorng to theore 4..6 f P s re of egree, then ( o ( P M o ( P B( o( P 5.. he Nuber M( 5... e M( s the rouct of stnct onc rreucble olynols (re of A of egree. - th hese olynols re the vsors of the Bernoull-Goss nuber (. B We hve: ( et F be rreuctble of egree such tht, F ves et α F( α ( α,,, s the sllest nteger such tht α α
6 ursh Journl of Anlyss n Nuber heory α α α ( α α α α α Becuse ( α α α α α α. be root of α α α hs roves tht : P ves eg or eg P P But α eg P. he revous le nswers the ueston: Wht re the res of egree vng the - th Bernoull-Goss nuber. e B?. ( o o P M P Concluson B ( o ( P If, there s exctly res of egree stsfyng the euton If, there s no re of egree stsfyng the euton. 5.. Nuber M( et P be re of egree whch ves M(, P s vsor of the -th Bernoull-Goss nuber B M ( [ ] M ( et α P( α ( α et:,,, n M α ( α α α α β α α, we hve : β β ( α α ( α α α α α α α α. here s two ossble cses: * Cse f β, then β β β therefore α s root of the olynol ( β, wth * β. We hve α α β α α β α β α β β α β α α β hen α α β α β α β β Moreover : α α β β β β β β β β f Snce * 4 5 ( s, s o. * * 5 (, 5 ( et F n rreucble of egree whch ves ( β F s of egree, becuse f δ s root of F, then δ δ β δ ( δ β δ β δ β δ δ β δ δ δ δ s root of hs roves tht: F ve eg F or eg F But δ eg F. herefore there s rreucble olynol of egree whch ves ( β ( β F ve M (. Concluson: s, s o, For n f F ve there s :. rreucble olynols of egree vng M( Inee, n ths cse, * 5 ( n therefore the euton hs two solutons n, β, β X X ( For ech β,,, there s rreucble olynols of egree whch ve ( β, n thus ve (M. hus, f P s n rreucble of egree whch ves ( β, α s root of P, then P( α.
ursh Journl of Anlyss n Nuber heory 7 herefore But: α α β α α β ( α α α ( α α M ( ( β β( β,, β β β β β β Snce β β,, β s root of (. hs roves tht : P ves M ( Cse f herefore: * β α β, then ( β β( β α α α α β β β β β β β β β β β β We set γ β β then r β β r ( γ r β r β, β α α n r s lner, Snce r ( β r α α Becuse: α α. So we hve: M ( o( P r α α Fro : ( β ( α et Q Irr( β,,, Q, hs egree n Q( b, becuse r β Q β β. β b Now we re loong for, Irr,,? β We loo for F( of egree such tht We hve : An then We set F,, β β β ( b β β F ( b F β n we wnt to get F [ ] we hve So we set ( F ( b ( ( b F ( F F ( F β ( F( F ( b ( b F b r β herefore the olynol s s follows Q( b hus Q( s n rreucble of egree wth constnt ter b, becuse we hve β n the other cse. Before conclung we wll nswer the followng ueston: for s there nfntely ny res P [] such tht : P o (
8 ursh Journl of Anlyss n Nuber heory 5... Prooston et, there s t lest one re P [] of egree such tht We cn ssue. P o eg M < Accorng to ([7], Prooston 5.5, we hve : l < N < where N ( s the nuber of rreucble olynols of egree [ ], l s the sllest re fctor of herefore If we h N ( > > P reer,eg M we woul hve :.e o ( P P eg M > ( < whch s ossble f. On the other hn: herefore N ( > ( N eg ( M > hus, there s t lest ( ( re of egree whch stsfy P o Concluson In ths er, we showe tht there re nfntely ny res P [] such tht References P o [] G. Anerson. og-algebrcty of wste A-Hronc Seres n Secl Vlues of -seres n Chrcterstc, J.Nuber heory 6(996, 65-9. [] B. Anglès n. eln. On Proble à l Kuer-Vnver for functon fels, to er n J.Nuber heory (. []. Crltz. An nlogue of the Bernoull olynols.ue Mth. J.,8:45-4, 94. [4] Ernst-Ulrch Geeler. On ower sus of olynols over fnte fels, J.Nuber heory (988, -6. [5]. Goss. Bsc Structures of Functon Fel Arthetc, Ergebnsse er Mthet un hrer Grenzgebete, vol.5, Srnger,Berln, 996. [6] Ireln K, Rosen M I. A clsscl ntroucton to oern nuber theory. New Yor: Srnger, 98. [7] M. Mgnotte. Algébre Concrete, Cours et exercces. [8] M. Rosen. Nuber theory n functon fels}. Srnger-Verlg, New Yor,. [9] J.. Shets. On the Renn hyothess for the Goss Zet functon for [], J Nuber heory 7(; (998, -57. []. hur. Zet esure ssocte to, [] J.Nuber heory 5(99, -7. [] Mohe Oul ouh Benough. Cors e Fonctons Cyclotoues, hèse octort e l'unversté e Cen, Frnce (.