Corps de décomposition de groupe de Galois PSL(2,7)

Texte intégral

(1)Corps de décomposition de groupe de Galois PSL(2,7) Annick Valibouze. To cite this version: Annick Valibouze. Corps de décomposition de groupe de Galois PSL(2,7). [Rapport de recherche] lip6.2005.001, LIP6. 2005. �hal-02545672�. HAL Id: hal-02545672 https://hal.archives-ouvertes.fr/hal-02545672 Submitted on 17 Apr 2020. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(6) . e1Lv®G>KNMSY[P^RVJSG½MSRVY>IQÏUl«GB[LZ\s[UVLOGXkY[Z{MSRTM#G>Z KG>UVKcZ1U:e\Z KY[P^]1Jre1LOe\bKY>à]_Y[J%RpMSRTY[I `¬G>RTJGB[LcKZ1IKY \Mj1RVLcI]\UVZ1JJ%Z\]kbcP%RVLcZ1Pª©Z1LhKLhª©Z1RLJ^Mj]1P^Y[]_Y[J%bhRVKcR zLcJOJ%Y>Z1JQºðs[P%Y[Z\]kLcJ}ºðM%PSG>I\J%RTM%RT7Jhe1L J%Y[IM DUyLcJ^M + LcM 7 = S7 LIMj6 J%R g 7T LcJ^M 1RVLcInKcY[I1I©Znª©Z1L Gal (f ) LJQMH}0ºðMSP%G>SI1J^7RTMSRpJ%R'LN7TMJ^4L, Z17TUTL5`, 7T = f /(x − α) k RVP%P^be1Z\KcMSRT1UVLvJ%Z1P k(α) Y α LJ^MZ1I\LP%G>KRTI1LOe1L f 'F#fJhUVG]1P%LcàRTfP%LvLN°5MSLcI1J%RTY[I I1Y[Z1J.]_Y[Z\[Y>I1Jye\Y[I1Ke1bcM%LP^àRTI1LP<ª©Z1LjUlmZ1Ie1LKLcJ.s[P%Y[Z\]kLcJLcJ^MY[ZI1Y[IUVLs>P%Y[Z1]_L e1LuG>UTY[RVJe1L f J^Z1P k +¸jY[Z1JjGB[Y[I1JUTLrUVL``LH]1UVZ\Jj]\P%bKcRVJJ^Z1RTXG>IM 5CÅ C ¯$ "!$#% &(' α )+*,.-0/+1 & *,32+, f 4 #5& Gal (f ) ,76 '98:;' -0/*<6 &('"& =?> 6@,BA % *DCE), % &('. '"&(G. '. k. %. >. %. + F = Galk (f ) 6 -?,76;F,71 ' ,BHI,B* '"&(7T G 4 , 7T5'O, N7T6 ) 7T7 =% S7 AJ,LK- )MF, > ,BHI,B* 2/*<6 S7 P AJ,QK- )@F, S1 × 6T1 Galk(α) (g) = Stab(F, 1) ,76 -7,76;F,71 +> + ' S1 × 6T7 S1 × 6T15 , S1 × 6T16 4. RTS. H. %. ' "' &V% w:G>P do5]_Y>MSd1fcJ%LXWK²dG>ª©Z1L¯s[P^Y[Z1]_L ]1G>P%`R 7T , 7T , 7T + LcM T *<6 U- / >bP^W* RT4 LcI©Mª©Z1L¯UTL¯s[P%Y[Z\]kL GÊZ1I1LG>KNMSRVY>I MS4PSGXI1J%5RpMSRT>L 6 J%Z\P. . zLnP%bcJ%Z1UpM²GXMaLJQMY[\M%LI©Z LcStab(T, I¶®G>RVJ%G>I1)MaG>s[RVP Stab(T, 1) J%Z\P¬UlmZ1I1RTª©Z1L ¨ lmLJ^MhZ\IµKG>JH]GXP^MSRTKZ1UTRVLcPe1ZKG>UVKcZ1U:e1LcJrs[P%Y>Z1]kLcJHe1LuhG>UVY>RVJ Y[P%1RpMSL e1LJ®GXKcMSLcZ1P%Je!l«Z\I1LHP%bcJ%Y[UpXG>IMSL#e1G>I1JUTL#KG>JYX¬U)GP^bJ^Y[UTXG>IMSLLJQM UVL]kY>UTo5I1q[`L g UVZ1R º`·`L|7>Y[RVPhÁ Y Â N Z RVI\J%RW:e1fJU)G¯]1P^L`RVfP^L ®G>KcM%Y[P%RTJSGXM%RVY[IÊe1L eG>I\J |7RmL> NW I1Y[Z1Je1RVJ^]kY>J%Y[I1Je!lmRVI\7Y>P%àGXMSRTY[I1JHJ^Z1PvJ^Y[IÎs[fP^Y[Z1]_Le1k(α) LuhG>UTY[RVJcx{fZ1]1=]_Y[(xJ^Y[I1−Jα)g e1Y[I1K ª5Z\L Gal (f ) J%Y>RTMH}ºðM%PSG>I1J^RTM%RT<LNMH]_Y[J%Y>I1J G = Stab(F, 1) WkUTLs>P%Y[Z1]_Lve\LuGXUVY[RTJ e1L g J%Z\P kk(α) +x5Y[RTM β Z1I1LhPSG>KcRVI1Lre\L g ¸jY[Z1JjGB[Y[I\JUTLrUVL``LHJ%Z1RpXG>I[M 5CÅ C a$ T$ \# ,BA % *]CE), G 6 % &(' -?,76;F,71 '^&(G ,BHI,B* ' S ×6T > S ×6T + > S ×6T +. 7T7 = S7 {2, . . . , 7} {2, . . . , 7}. %. 1 1 1 7 15 > = ' & S '^&"a 1 S − β) k(α)(β) B 6 + ) )% S' 1 × 6T16'^&(G AJ,762+,V' K- 6_Nb2+& ,76 /+1 ,B)`-?6 = -B- 2' )1 AJ&,76[S 2+, & g/(x > NfeT= ' &(: 6 S * & -?,76;F' ,71 ,BHI,B* Si15h > 4",E4c% CE),2+,76 /+1 ,B)+-?6dA * / -?,76 P 1, 4 /+1 ,B)`-gA * / -7,3, )+*D2+,32+,VK- P 5 ) 5 4 RTS % ' '"&V% wDY[Z\PUTLJhs[P%Y>Z1]kLcJ}ºðM%PSG>I1J^RTM%RT7J I1Y[Z1JGB[Y>I1J7Y[P»º + LNM Kb`H LI*<Mj6 {Y -U/ w3Y[Z1*WP4 UVLrs[P^Y[Z1]_L S × 6T e!lmY[P%e1P^L\k W g6TJ^15Lr®G>KcM%6T Y[P%RT16J%LrI1bcKLJ^JSG>RTP%LcàLcI©M KY[`]1UVfNMSLcàLcI©MyeG>I\J UlmLN°5MSLcI1J%1RTY[I 1 e1Le1Ls[P^bjvk J^Z1P \w3Y[Z1P + W. UVLKBG>UTKZ1U\e1LJKBG>Pê1RVI1G>Z{°e1LcJyY>P%1RpMSk[x]/ LJ.e1L<U~lG>g(x) KNMSRTY[I>e1L Stab(6T , 1) J%kZ1P 7 e1Y[I1I1L LcMjÑ1 ¸Y[Z1JLcIne1bce1Z1RTJ%Y[I1Jjª©Z1LQ M l5mnl_p@ o7@ o"lWq $ sr$#% & ,B* ' α , ' β 2+,B)td-U/+1 & *,76.2 & 6 '"& *1 ' ,76.2+, f 4vu. 6T7 {3, . . . , 7} . % G% )6O/ *<6 ' & ' ' & 'w& S "' &"a Galk (f ) = 7T5 6 = , 'f6@% ,B)`& AJ,BHI,B* 6 g = f /(x −= α)' ,76 -- & 2)S 1 & JA ,x' 6B)+- k(α) = ' , CE)5& , g/(x @ 6 , + / 1 @ 6 y , B 6 + ) B , * ` ) * + / 1 B , ` ) A * / 7 _ , , ) ` * /+1 ,B)`− β) k(α, β) h S '^&"a Sch 4 l -- 2)1 AJ,32+,.2+,VK-. \Ð _zÈDË{¥'¤3Ë£:¤¶¥'¢:¡{+É£:Ìµ£}|+¥'¢~_{+¢<É¦) ¦§¢< FHG>I1JjKLNM%MSLh]GXP^MSRTL>W1I\Y[Z1JJ%Z1]\]kY[J^Y[I1Jjª©Z1LhJ%Y[IMKBG>UTKZ1UTbJ_ | Gal (f ) = 7T k. 5.

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(8) ` B ( W ` X B . |}[UVL ®G>KcM%LZ1PaUVRTI1bBG>RTP%L. . LcM¬UVL®G>KcM%LZ1Pe1Lne1Ls[P^biÑ l(α, β, x) e\L J^Z1P |7>Y[RVPw¹P^Y[]kY>J%RTM%RVY[I }5Ð zLcJn`bcM%d1Y5e1LJ]_Y[Z1P ª©Z1LµU)GÊKY[I\e1RTM%RVY[I | ¬J%Y[RpMJSGXM%RVJQ®G>RTM%L¯J%Y[IMI1Y[Ò\P%LZ\J%LJ |7[Y>RVPW+]GXPL°\Lcà]\UVL>W3Á k Â Nk¸Y[Z1Jj]_Y[Z\[Y>I1JHZ\M%RVUVRTJ%LcPUVLJjP%bcJ%Z1UpM²GXM%Je\Z ]G>P%G>s[PSGX]1d1L ]1P%bcKbe\LIM ª©Z1R7Y[Z1PÎ1RVJ^J%LcI©MyG>Z\J%J%R+UTLJ<®G>KNMSLZ\P%J h LcM l e1L g ©x5R Gal (f ) LJQMe1bcM%LPQº àRTI1b>WRTU'LJ^M#]kY>J%J%RT1UVLhe!lmY[\M%LI1RTP h LNM l GB>LKve\LJ#KG>UVKcZ1UVJ pº»G>e1RTª©Z1kLJj]1Z1RTJ%ª©Z1LvUTLJ e1Ls[P^bJhLIµK²dG>ª©Z1LXG>P%RVG>1UTLvJ%Y>I©MKcY[I1I©Z1Ja|7>Y[RVPÁ t Â N ¹I ! "" WeG>I\JZ\I1LKY[`º ÒZ1I1RTKBGXM%RVY[In]1P%Rp[bLXW!À$ #&Y %>Y0oG>àG¬]1P^Y[]_Y[JSGe1LvKBG>UTKZ1UTLPje1RVP^LKcM%L`LIMrUTLvKY>P%]1J e1Le\bKY>à]_Y[J%RpMSRTY[Ie1L f GB[LcKe\LJyKBG>UTKZ1UTJ pº»G>e\RVª©Z1LJ.LcIa7Y[P^`¬G>IM.UVGhàGXM%P%RVKcLe1LJ KY5LcgKRTLIMSJHe\LvUlmRVe\bBG>UàG½°{RVàG>U!K²d1LcP%K²d1bX'wDY[Z\PHkY>P%I1LcP#JSGa`¬GXM%P%RTKL>WRTUZ\MSRTUVRVJ%G>RTM U)Gv7Y[P^ÒZ1UVLe1LÁ ' Â ª©Z1R7Y>Z1P%I1RpMUVLcJ e1Lcs[P%bcJje\LJ`Y[I1q[`LJ RTI1RTM%R)G>Z5°e1LrUlmLI1J^LÒ\UVL MSP%RVG>I1s[Z\U)G>RTP%LLcI1s[LI\e1PSG>IMaUlmRVe1bG>U( ¹I1KY[P^LnZ\I1L7Y>RVJW:U)G¯àbNMSd1Y5e1L]1P^Y[]kY>J%bLRVKcR LJ^M ]1UVZ1JyLcgKBGXKLHKG>PLcUVUVLje1RTJ%]_LI1J^L#e1LHKG>UVKcZ1UVLcPy_Y[IaI1Y[Ò1P^L#e1L#KLJKY5LcgKRTLIMSJ |®J%LcZ1UVLcJKcLPQM²G>RTI1LJP%LcU)GXM%RVY[I\J J%LcP%Y[IMÏKBG>UTKZ1UTLP² x{Y[RVLcIM e1LJiPSG>KcRVI1LcJne1RTJ^M%RVI1KNMSLJne1L f #w3Y[J%Y>I1J α = (α , . . . , α ) 1 7 z.lmRVe\bBG>U α1, . . . , α7 h(α, β, x) f (x)/(x − α)(x − β) k(α, β). 1 , . . . , x7 ] | r(α1 , . . . , α7 ) = 0} L J^MZ1I RVe\bBG>UHàM G½°{RV=àG>{rUje∈G>I1k[x J k[x KBG>PI1Y0oGXZ e1Z³`Y[P%]\d1RVJ^àL J^Z1P*)^LcKNº , ..., x7 ] MSRTe!lmbcXG>UTZGXMSRTY[I³LIMSP^LEU~lGXI1I1LBGXZ 1k[x LNMiUTL¯KY[P^]1Je1LEe\bKY>à]_Y[J%RpMSRTY[I , ..., x ] 1 7 : '"&V% UDLJ^M e\L ¨ LcMhRVe\bBG>U3LJQMG>]1]_LUVbvUl & S *<6 K α7Lc)I1J%LcÒf1UTLhM%P%RVG>I1s[Z1UVG>RVP^L#e1Lh]_Y[UTo5I1q>2 à/LcA}J 2+,76 α -?,BA / LI1s[=LcI1k(α e1P^b1], GX. .P.Z1, In. r (x ), r (x , x ), . . . , r (x , . . . , x ). 1 1 2 1 2 7 1 7 XY K²dG>ª©Z1Lr]_Y[UTo5I1q>àL LcJ^MZ1I\RTM²GXRVP%Le1LHe1Lcs[P%b LI \z'LHs[P%Y>Z1]kLe1LHe1bcKY[`º di e1LcJvx]_iLP^ÒZ\M²G½MSRVY>I1Jhe1L ]kY[J^RTM%RVY[I Dec(M) e1LaKrLci MvRVe1bG>U|®R~«LX'UlmLI1J^LÒ1UTL¬ ª©Z1R U)G>RTJ%J%LcIM M RVIXG>P%RVG>IM LJQM#Z\IiKY>I+)^Z1s[Z1be1L Gal (f ) _UJclG>]\]kLcUVUVLhUTL K- % S)@F7 ,T2+, , % & ©U_LcJ^MyRVJ^Y[àY>P%]1d1LjG>Zs[P^Y[Z1]_Lke1LcJ kº»G>Z\M%Y[`Y[P%]1d\RVJ%`LJ¹e\L K / A _ 6 + 2 , α B 6 + ) k x{Y[IY[P%e1P^L k"'OLJQM#UTLr]1P%Y5e1Z1RpM d · · · d |7[Y>RVPhÁ Â ]kY>Z1P MSY[Z{MKLhª©Z1R]1P%bcKfe\L0N wDY[Z1PKG>UVKcZ1UVLcP UVLrKcY[P%]1J K W1I\Y[Z1J1 e1Lc>Y[7I1Jje1Y[I1Khe1bNMSLcP%`RVI1LcP Z1IMSLURVe1bG>U M ]G>P Z1InLcI1J%LcÒ1UTLhM%P%RVG>I1s[Z1UVG>RVP^L#U~l«LcI1s[LI\e1PSG>IM zLUVL``LyJ^Z1RTXG>IM¹e1bcKY[Z\UVL e1LcJ¹P%bcJ%Z1UpM²GXM%J<e1LrÁ Â àG>RTJDJ%GHe1b`Y[I1JQMSPSG½MSRVY>Ive1RTP%LKNMSL I!l«LcJ^M]G>JKY>à]1UTRVª©Z1bcL> 5CÅ C r^!c#% &(' M )`* & 2 S /AH /pt & H /A2+,-?,BA / '^&V% *<6E4.-xA % -?6 | A0/ ,B*<6@,BH a AJ,2+,76 & 2 S /) tgH /pt & H /) t]2+,[-?,BA / '"&V% *<6T,76 '=p% -BH S 2+,76 σ.M %+1. % ' <F /-?1 )`- Sn 4 |}[ 12</CE)5, & 2 S /A σ.M ,76 ' A0/ & 2 S /A 2+,76 σ−1 .α : -?,BA / '"&V% *<6c2+,\K- % )@F,c2+,$2 S 1 % H : % &('^&V% N F 6 * Dec(σ.M) = Dec(M)σ = σ Dec(M)σ −1 P 4 Ml5mnl_@p7o @o"lWq rT$ 43 * 6 % )6 : K- % )@F, H 2+, S ,76 ' AJ,cK- % )@F,2+,2 S 1 % HxF % 6 &('"&V% * & S & '"&V% n ' > > 25' / )`* 2 /A`& H /p' t H /A hr1 , r' 2 , & . . . , r7 i' 2+,i-?,B% A / *<6 S ,BAWCE), rG 1 S =& 8 f r2':= 9 g r3 = h , r4 = l 6 , 6@,B)ÀJ,BHI,B* 6 H ,76 )+*1 *76)XK) 2+, 7T5 - /* | Stab(H, 1) ⊂ S × S , ' |}[ Stab(Stab(H, 1),1 2) ⊂6 S × S × S × S 4 1 1 1 4 σ.

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(10) . x{Y[RpM M Z1IRVe\bBG>Uk`¬G½°{RT`¬G>U+e1LrP^LUVGXMSRTY[I1JyM%LU!ª©Z1L r = f W r = W 'DY[Z{Mve!l«G>_Y[P%eW'UVLs[P%Y>Z1]kL Dec(M) LJ^MZ1IEK1Y[I+)^Z\s[Z1b2 e1L g r3|7>= h r = l 4 Y[RVP.z'Lcà`L Ð\ N[¸Y[Z1J.J%G0>Y[I1Jª©Z1L σ ∈ Dec(M) J%RLNM.J%LcZ1UVLcàLcI©MJ^R]kY[Z\P 7T 5 MSY[Z\M%LP^LUVGXMSRTY[I W ¨ LKR3LJQM#e\Y[I1K>W_LIn]G>PQMSRVKcZ1UVRTLPcW , . . . , ασ(7) )=0 + { x R MSLcU!ª©Z1L σ(1) = 1 G>UVY[P^J [bP^RTbh]_Y[Z1PUTLJrP%∈LcU)GXMM%RVY[I1r(α J rσ(1) σ ∈ Dec(M) 1 , r2 , r3 , r4 <F#Y[I1K JSG½MSRVJQ®G>RTM 0| # r2 (α1 , αJ%σ(2) )M = g(α1 , ασ(2) )W!e1L]1UTZ1JDec(M) JrPSG>,KcαRVI1σ(2)LcJr)e1=L g(α [ Y I { x ~ R W W'G>UTY[P%J =KG>rPr2(αUVLcσ(1) α2 , . . . , α7Wj]_Y[Z1P σ(2) = 2 1 , x) ¸Y[Z1JGB[Y>I1J 0 = ri (ασ(1) , ασ(2)LcJ^, MO , αU~l«σ(i) )RVª©Z1=LPSrG>i (α i JQM = 3,Lc4M 1 , α2 , ασ(i) ) B K > G P \ Z 1 I c K V R 1 I L 1 e L L σ(3) = 3 α3KBG>PUVLcJbcUVbcàLcI©M%Je\L h(α α3 σ(4) ∈_F#EY[I1=K 1 ,JαP%2G>,Kx) % J [ Y I M V U c L T R 1 I L j J 1 e L {α4 , α5 , αJ%6GX,MSαRTJ^7®}G>RpM|}[ E l(α1 , α2 , x) Dec(M) jbcKRV]\P%Y5ª5Z\L`LIMhJ%R LcJ^MhZ1I¯KcY[+I )^Z1s>Z1be1L G>UVY[P^J#RTU:LJQMhUVLOs>P%Y[Z1]_LOe1Le\bNº KY[`]kY>J%RTM%RVY[Ie!lmZ1IRVe\HbBG>U5`¬G°\RT`¬GXU M |7[Y>RVP<z'Lc7Tà5`L Ð{ x{Y>RTM {r , . . . , r } UlmLI{º J%L`v1UVLvM%P%RVG>I1s[Z1UVG>RVP^LrLI1s>LI1e1P%G>IM M Jl«G>I1I©Z1UVG>IMrLI α , . . . , α '1x5R H J%7GXMSRTJ^®G>RpM UVLJKY[I1e\RTMSRTY[I1Jh| yLNM|}[yG>UVY[P^JW{e!lGX]1P%fcJhÁ ' Â W\UTLJ e\Ls[P^bJ1 P%LcJ%]_LKcM%7RT7Je1L r , . . . , r LI x WTc x J%Y[IMvt{W k\W W«Ñ1W W LcM |®KXl«LcJ^MZ1IEJ^RV`]1UVLvKBG>UTKZ1U<e\LJQM²G>\RVUVRTJS1GXM%LZ1P^J²7 ¨ Y[à1`L M7 LJ^MaàG½°{RVàG>UW3I1bKcLJ^JSG>RTP%L`LIM r LcJ^MaRVP%P^be1Z\KcMSRT1UVL J^Z1P k W r J%Z1P W r LcM r J%Z\P k(α , α ) Z [LKjUVLcJ.P%bJ^Z1UTMSGX1MSJ.e1Z]G>P%G>s[P%G>]1d1L } WI\Y[Z1JGB2[Y[I1J k(α ) 1 e1Y[I1K r 3= f W 4r = g |7UVL1 J%LcZ12 U_]kY[Upo5I1q[`Lje\Le1Lcs[P%$b kJ^Z1P k(α )W r = h |®UTLJ^LZ1U ]kY[Upo5I1q[1`LHUVRTI1bBG>2 RTP%LJ%Z1P k(α , α )LcM r = l |®UVL#J%LZ\U']kY[Upo5I1q[1`LHe13Lhe1Ls>P%brÑJ%Z1P 1 2 4 k(α1 , α2 ) ¨ Y[à`L<eGXI1J'U)G ]1P^Y[]_Y[J%RpMSRVY>IH]1P%bcKbce1LIMSLXWK²d1Y[RVJ^RVJ^J%Y[I1J'ª©Z1L.UTLJ!P%G>KRTI1LJ MSLUTUVLcJrª5Zl«Y[Iµ]1Z1RTJ%J^L]_Y[J^LP r = f W r = g W r = h LNM r = l 'zG J%Y>αà1 ,`. .L. e1, αLJ 7 PSG>KcRVI1LcJyLcJ^MyY[]1]_Y[J%bcLHG>ZKY5Lcg1 KRTLIM λ 2e\L x6 e1G>3I1J f ©¸Y[Z\J4 ]_Y[Z\[Y>I1Jye\Y[I1Kj]kY>J%LP ¨ Y>à`LU~lGXI1I1Y[I1KcLUTLjM%d1bY>P%f`L#J^Z1RTXG>IMW{I1Y>Z1J e1RTJ%]_Y[I{º rJ%Y[7 I1=Jjx 1 + · · · + x7 + λ e1b )%Ïe1LhMSY>Z\MSLcJUTLJRVI\7Y>P%àGXMSRTY[I1J]_Y[Z1P]1P^bNºðKBG>UTKZ1UTLPUVLcJP%LUVGXMSRTY[I1J LcM P%LJ^]kLcKcM%RT[LcàLcIM UVRTI1bBGXRVP%LcJ¹LI x LcM x |§[Y[RTP¹e1b`Y[I1JQMSP%GXMSRTY[Iae\LjUVG]\P%Y[]_Y[rJ%5RpMSRTY[I+rN6 5 6 +? l+5 C r srT#% &(' f )+* F % A * Hg, )+* &(G /- & S 2+,LK- % )MF, 2+, , /A % & 6 7T RTS. H. %. ' '"&V% *<6 U- / LcM *W4. #% &(' % S 5 6B)`- k 4 '"& = & H ' = hσ = + ) * + 2 7 , 6 1 7 * 6 X ) K ) (1, 4, 3, 5, 6, 7, 2), (2, 5)(3, 4)i % &('"&V% Nfe ']Nb8 % % &('"&V% 8 % 6L2+, 7T' 5 6E/ 6 % / 6E/* AJ,7S 6 % 1 *W% 2 &('^&V% *<6 , + 2 L , A / W F F 6 *. 4 4 x A -76 a H P & S P & S ',76 & AJ,[K- & )@F,Q9 2+,O2 1 HxF 6 * 2+,3A0/ 2 /A H /pt H /A,B*+K+,B*W2- F</-3A0/ ,B*<6@,BH AJ, - /*+K)+A / -7, +% 1 > & S -B6B)`6B)`RTS. { f (x1 ), g(x1 , g2 ), h(x1 , x2 , x3 ), l(x1 , x2 , x4 ), h(x4 , x1 , x5 ), h(x3 , x1 , x6 ), h(x5 , x4 , x7 ) }. .. % ' & & '^& ' > ' F )+'^&"- a α , β 2+S

(11) ,B) t -U/+1 *,76 > 2 6 *1 ,76 2+', f = g(α, ' x)& =S & f /(x − α) ,76 2 )1 AJ, +2 ,y ' 2+,VK- 6B)+- k(α) ' = h(α, ' β,& x) S ,76 AJ'", &"a /+1 ,B)`-\A * Sj/ h -7, 2+, g/(x−β) 2+, g/(x − β) k(α, β) , l(α, β, x) ,76 AJ, /+1 ,B)+- -- 2)1 AJ, 2+, 2+,VKk(α, β) 4 % ' '"&V% Fvl«G>]1P%fcJOU)G]1P%Y>]kY[J^RTM%RVY[I Ð\} W LJQMUVLs[P%Y>Z1]kLae1Le1bcKY[`]_YXº H J%RTM%RVH Y[I *<e!6 lmZ1-UI/ RTe1*WbB4 G>UhàG½°{RVàG>U LcM l(x 3 ),UTL >G ]1]G>PQMSRTLI1I1LcIMÏ¶UlmLI\J%L`v1UVLÎMM%P%RVG>I1s[fZ1(xUVG>RV1P^),L¯g(x UlmLI11,s>gL2I1),e1P%h(x G>IMB1#, xF#2Y[, I1xKÊ Jn1M%,P%Y[xRT2Ji, xP^4LN)º U)GXM%RVY[I1J h(x , x , x ) = σ.h(x , x , x ) W h(x , x , x ) = σ2.h(x , x , x ) LcM 4. 1. 5. 1. 2. 3. 3. 1. 6. 1. 2. 3.

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(13) ` B ( W ` X B . >G ]\]G>P^M%RVLcI1I1LIM#G>Z1J^J%RDÏ M ¨ Y>à`LhLUVUTLJJ%Y>I©M UVRVI\bBG>RTP%LJLI W LNM P%LcJ%]_LKNMSRT>L`LIMBW+I1Y>Z1JY[\MSLcI1Y[I1JjUVLrP^bJ^Z1UTMSGXMB Ñ1_ z#¢ ~_{XÈD¡'È:¦§É¢<3É .IÊZ{MSRVUTRVJ%G>IMhU)Gn`bcM%d1Y5e1LQÏnU~lGB[LcZ1s[UTBL ae1La®GXKcMSY>P%RVJ%GXMSRTY[IeG>I1JvUVLJLN°5M%LI1J^RVY[I1J ]kY[Z\PyKG>UVKcZ1UVLcP¹UVLJ.P^LU)G½MSRVY>I1J r LNM r W©RVU1®G>Z\M.®G>KNMSY[P^RVJ^LP.UVL]_Y[UTo5I1q>àL l(α , α , x) 5 6 1 2 eG>I1JUlmLN°5M%LI1J^RVY[I k(α , α , x)/ e\Le\Ls[P^b k"'J%Z1P k < l(x) > 1 2 .IZ\M%RVUTRVJSGXI©MyUVLcJ KBGXUVKZ\UVJ º»G>e1RTª©Z1LJcW1À #"Y %XY0oG>àGOK²d1LP^K²dG>RpMUVLJ]kY[Upo5I1q[`LJ r5 LcM r +F#Lh]1UTZ1JWI\Y[Z1JJSGB[pY>I1Jª©Z1LXW+]kY[Z\P i = 3, 4 W º r 6∈ k[x , x , x ] GB>LK deg (r ) < deg (r ) = 7 W deg (r ) < deg (r ) = 6 i 1 2 x i x 2 º deg W ideg (r ) =x 0 iLNM deg (rx ) 1= 4 (r ) = 1 Z RVI\J%RxWJ%R 3 LJ^xMKcY[4I1I©Z!WUVLcJr]kY[Upo5xI1q[`4 LJ r LcM r ]_Y[Z1P^P%Y[IMh·NMSP%LY[{MSLI©Z1J Gal (f ) k ]G>Pe1LJKG>UVKcZ1UVJ pº»G>e1RTª©Z1LJJ%G>I1JG0>Y[RVPÏKG>UVKcZ1UVLc3Pe1LJ4KY5LcgKcRVLIM%JJ^Z1]kLcP Z\J 3lWq3 ~A3p@ o lWq zG`bcMSd\Y{e1Le1bKcP%RpMSLOeGXI1JrKLNMhG>P^M%RVKcUVLvLcJ^MhGX]1]1UVRTKBG>\UVLeG>I1JHM%Y[Z\M Jô5J^M%f`Lne1LKG>UVKcZ1U7Y[P^àLcU .UTUVLJclGB>fP%LnIGXM%Z1P%LcUVUVLcàLcIM]1UVZ\JLcgKBG>KcLnª©Z1LUTLJ G>Z\MSP^LJàbNMSd1Y5e1LcJyKcY[I1I©Z1LJÏKcL )^Y>Z1P©w3Y[Z1P·cMSP^Ls>bI1bcPSG>UTRVJ%bcL>WLcUVUVLe\LàG>I1e1LjZ1I1L bcMSZ\e1LvGX]1]1P%YX7Y[I1e1RTLHLcMMSP^Y[]KY[`]1UVRTª©Z1bLH]_Y[Z1P·NMSP^Le\bc[LcUVY[]\]kbcLRTKR. h(x5 , x4 , x7 ) = σ 3 .h(x1 , x2 , x3 ) x5 x6 x7. 1. 3. 3. 1. 2. 2. 4.

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