MeBo 4.9. 6:9 fz z ±, ± z \ ±, ± z z 4 C gz sl z dz C z 4 f sl z 4 [,, [,, [,, [, D C g + O A B B g sl sl g g g z g g z 4.. z dz ϖ ϖ z 4 g z 4 dz z 4

For Tutors MeBo Edted by ame

MeBo 4.9. 6:9 fz z ±, ± z \ ±, ± z z 4 C gz sl z dz C z 4 f sl z 4 [,, [,, [,, [, D C g + O A B B g sl sl g g g z g g z 4.. z dz ϖ ϖ z 4 g z 4 dz z 4 g g ϖ, g g ϖ dr r 4 fz D r > fr + r 4 dr ϖ, D r4 dz AB + + z 4 dz r4 dr s 4 s ds s ds ϖ

g + ϖ + ϖ O B+ + z 4 A ϖ A 9 ϖ + ϖ r > fr r4 g ϖ ϖ A 9 D ± ϖ ± ϖ 4 sl p d p C d ϖ p d B Aϖ p d D B C p D sl ± ϖ ± ϖ 4 D : AB AB [, ] A BCDB B C D B sl sl p d p 8 slz slz mϖ + nϖ m, n slz + ϖ slz, slz + ϖ slz 3 slz mϖ + nϖ m, n 4 slz m + ϖ + n + ϖ m, n slz sl m + ϖ + nϖ,

MeBo 4.9. 6:9 3 sl m ϖ + nϖ, sl mϖ + n + m, n ϖ, sl mϖ + n ϖ L {mϖ + nϖ m, n } L C sl Remann R C/L O, Aϖ, B3ϖ + ϖ, Cϖ + ϖ R OABC BC, CD C { } C sl : R C R C ±, ± C 4 : 3 slz m + ϖ + n + ϖ m, n m, n m, n m n z < Rez < ϖ, < Imz < ϖ w slz Rew >, Imw > w z w slz w ++ w dt w t 4 + ds w s 4 4 s ds w + s + ds s 4 + sl + sl + sl w w sl + ϖ w + ϖ sl z ϖ ϖ s 4 ds ϖ + ϖ s4 ds sl z w t 4 dt w + z slz Taylor slz z + w z ϖ ϖ + w slz sl z ϖ ϖ z ϖ slz sl z ϖ ϖ ϖ + slzsl z ϖ ϖ c z ϖ ϖ c sl sl ϖ slz sl z ϖ ϖ

4 4 sl z ϖ ϖ slz ϖ sl 4 + ϖ 4 sl + + + + dt t 4 dr + r 4 + ϖ ϖ clz sl z clz clz mϖ + nϖ m, n clz + ϖ clz, clz + ϖ clz 3 clz m + 4 clz mϖ + n + ϖ + nϖ m, n ϖ m, n clz cl mϖ + nϖ, cl m + ϖ + nϖ, cl m + ϖ + n + ϖ, cl m ϖ + n + ϖ m, n slz, clz + + + + Re > Im > Re < Im > Re < Im < Re > Im <

MeBo 4.9. 6:9 slz, clz slz ϖ + O ϖ clz ϖ+ O ϖ O

6 3 ϖ, ϖ π C G ϖ z z + ϖ ϖ z z + ϖ π z zg slz clz slz, clz ϖ, ϖ, ϖ + ϖ G L L slz, clz G/L 3 8 G slz, clz sl z + ϖ + ϖ G slz, slz fz z slz, slz, slz, slz z slz, f a a ±, ± aclz, a clz slz, slz, slz G 3 z slz clz z slz clz z slz clz z slz clz z + ϖ + ϖ slz clz z + ϖ slz clz z + ϖ slz clz z + ϖ + ϖ slz clz z + ϖ slz clz z + ϖ slz clz z + ϖ z + ϖ clz clz slz slz 4 slz, clz Hurwtz snz, cnz slz, clz v mϖ + nϖ m, n u fu sluslu + v gu cluclu + v fu, gu ϖ, ϖ fu + ϖ slu + ϖslu + ϖ + v { slu}{ slu + v} sluslu + v fu {z ϵ < Rez < ϖ ϵ, ϵ < Imz < ϖ ϵ} ϵ

MeBo 4.9. 6:9 4 slz, clz 7 fu ϖ + ϖ, ϖ + ϖ v gu ϖ, ϖ v 4 gu + clv fu + clv ϖ + ϖ, ϖ + ϖ v ϖ, ϖ v ϖ g + ϖ ϖ cl + ϖ ϖ cl + ϖ + v ϖ g + ϖ ϖ v cl + ϖ ϖ v cl + ϖ cl v clv clv clv, gu + cl v ϖ + ϖ, ϖ + ϖ v ϖ ϖ ϖ f sl sl + v clv ϖ ϖ ϖ clv, f v sl v sl cl v clv fu + clv ϖ, ϖ v 4 fu + clv gu gu + clv ϖ gu + clv + cl v gu + clv ϖ, v fu + clv ϖ + ϖ, ϖ + ϖ v fu + clv gu + clv sluslu + v + clv cluclu + v + clv u sluslu + v + cluclu + v + hv u clv clv hv slslv + clclv + clv + + clv clv clv clv cl v 4 3 clx + y slx + y clxcly slxsly + slxclxslycly slxcly + clxsly slxclxslycly

8 fu + clv fugu + clv gu + clv fu + gu clv fu + gu fugu + cl v sluslu + v + cluclu + v sluslu + vcluclu + v v x + y, u x clx + y sl xsly + cl xcly sl xslycl xcly slxsly + clxcly + slxslyclxcly x ϖ + x ϖ cl + x + y slx + y, ϖ ϖ sl + x sly + cl + x cly ϖ ϖ + sl + x slycl + x cly clxsly slxcly + clxsly{ slx}cly clxsly + slxcly clxslyslxcly cl u sluclu + v slv cluslu+v+ slv sluclu + v cluslu + v + slv slv sl v slv sluclu + v + cluslu + v + sluclu + vcluslu + v v x+y, u x

MeBo 4.9. 6:9 9 Jacob snz, cnz, dnz < < fz z ±, ± z z z \ ±, ± z C gz sn z z z dz C f sn z [,, [, D C g + O A B C B C g sn sn g g g z z z g K g z z.. z z z dz fz D < r < fr + r r AB + + z z dz dz K r r dr

Jacob K K r r dr g + K + K, g K K BC < r fr + + + r r z z dz + r r dr s s ds K. s s s ds g + K + K K K O C+ + z z A K A 9 B K + K 9 K AB C K sn w K + + z z dz + s + s ds K r r dr + s r r : s : + s + s + s ds r + s + s s + s, r + s + s + s, dr s + s + s s + s + s + s ds s + s + s s + s + s + s ds s + s + s + s ds r r dr + s s + s + s + s ds s + s + s + s ds

MeBo 4.9. 6:9 snz, cnz, dnz snz, cnz, dnz < <, K t t dt, K t t dt snz K K + + K + O K + + + snz 4K, K snz + K snz, snz + K + K snz, snz + K snz { z mk + nk sn mk + nk m mod m mod v snz mk + n + K m, n m mod m mod v snz sn ± 4m ± K + nk, sn ± 4m ± K + n + K m, n v Taylor Laurent 3 z snz z + 6 z K snz + z K + K snz z K snz z 3 + 4 + 4 + z K + + 6 z + z K + 4 6 + 4 z K K 4 6 + 4 z K 4 + z K + 74 + 7 36 z K K 4 + z K 3 +

Jacob cnz sn z K O K cnz 4K, K + K cnz + K cnz, cnz + K cnz, cnz + K + K cnz z m + K + nk { cn m + K + nk m n mod m n mod v cnz mk + n + K m, n m n mod m n mod v cnz cn mk + nk m n mod, cn mk +nk m n mod, cn m+k +n+k m n mod, cn m + K + n + K m n mod v Taylor Laurent z cnz z + 4 + z 4 + 4 z K cnz { z K + z K + K z K cnz cnz { z K 3 + 64 6 + 6 { + z K K 4 4 z K 6 z K 84 8 7 36 } z K } z K K 4 + } z K 3 +

MeBo 4.9. 6:9 snz, cnz, dnz 3 3 dnz sn z K O K dnz K, 4K dnz + K dnz, dnz + K + K dnz, dnz + Kdnz z m + K + n + K { dn m + K + n + K n mod n mod v dnz mk + n + K m, n n mod n mod v dnz dn mk + 4nK, dn mk + 4n + K, dn m K + 4nK, dn m K + 4n + K v Taylor Laurent z z 4 + z K dnz { + z K + 4 4 4 z K + K dnz { z K K 6 z K dnz z + 4 + 4 4 { dnz z K + 6 } z K 4 z K K 3 + 4 6 + 6 z K + 74 + 8 8 36 } z K K } z K 3 +

4 Jacob 3 snz, cnz, dnz z mk + n + K snz, cnz, dnz 4K, 4K cnz snz, dnz cnz, snz dnz snz, cnz, dnz mk + nk snz, cnz, dnz, K, K + K, K K K + K K snz cnz dnz snz cnz dnz snz cnz cnz snz snz dnz dnz snz cnz dnz dnz cnz snz snz + K dnz dnz + K snz cnz dnz + K + K snz dnz dnz cnz + K cnz dnz snz + K dnz cnz cnz + K + K cnz snz dnz + K snz cnz + K cnz snz + K + K

MeBo 4.9. 6:9 4 K, K 4 K, K z snz cnz dnz z snz cnz dnz z + 4K snz cnz dnz z + 4K snz cnz dnz z + K snz cnz dnz z + K snz cnz dnz z + K + K I snz cnz dnz z + K z + z + K + K cnz dnz snz dnz cnz snz dnz dnz snz cnz dnz cnz snz snz cnz snx, cnx, dnx u snusnu + v K, K u K u K v K snz K snk + v snv snv K v snk v snz K sn v snv 3, 4 cnucnu + v, dnudnu + v u K, K u K u K v u K u K u snusnu + v snv snv cnucnu + v dnv snv dnudnu + v cnv snv dnv snv cnv snv cnucnu + v + snusnu + vdnv Remann u cnucnu + v + snusnu + vdnv fv u hv cnv cnucnu + v + snusnu + vdnv cnv v x + y, u x cnxcny snxsnydnx + y cnx + y dnudnu + v + snusnu + vcnv u dnudnu + v + snusnu + vcnv gv u gv dnv dnudnu + v + snusnu + vcnv dnv v x + y, u x dnxdny snxsnycnx + y dnx + y

6 Jacob, snxsny snxsny cnx + y dnx + y cnxcny dnxdny cnx + y dnx + y snxsny cnxcny snxsny dnxdny snxsny cnxcny sn xsn y snxsny dnxdny cnxcny snxsnydnxdny sn xsn y dnxdny snxsnycnxcny cn, dn cnx + y cnxcny snxsnydnxdny sn xsn y dnx + y dnxdny snxsnycnxcny sn xsn y sn snu + v cnv cnucnu + v snudnv cnv sn usn v cnucnucnv snusnvdnudnv snudnv sn usn v cnv cn u cnvsn usn v + cnusnusnvdnudnv snudnv sn usn v cnvsn u cnvsn usn v + cnusnusnvdnudnv snudnv sn usn v cnvsnu cnvsnusn v + cnusnvdnudnv dnv sn usn v cnvsnu sn v + cnusnvdnudnv dnv sn usn v cnvsnudn v + cnusnvdnudnv dnv sn usn v cnvsnudnv + cnusnvdnu sn usn v snx + y snxcnydny + snycnxdnx sn xsn y

MeBo 4.9. 6:9 7 3 P O A r cos θ ϖ OP ϖ P OP OP z, OP w w slz ϖ sl slz slz slz slz slz slz sl 8 z sl 4 z + sl 6 z + sl z 6sl 8 z sl 4 z + sl 8 z sl 4 z + sl 6 z sl z 6sl 8 z + sl 4 z + slz slz z 4 sl 4 z α, α, α, α slα, slα slα, sl α slα, sl α slα ϖ slz sl + ϖ z slz 4 8 6 slz s, clz t Remann C/L E : s t E n E n {slα, clα α C/L n } GalQ/Q C/L Z /Z F F AutZ /Z GL, F Z /Z, 4 GL, F E 4 slz C/L L L slz clz Z /Z Z[]/ Z[]/ Z[]/ + F F

8 3 GalQ/Q[] Z[]/ Z[]/ + GL, F Z[]/ + Z[]/ GalQ/Q[] a a,, 3, 4 4 a a,, 3, 4 4 Q[] 8 Q 6 a b a \, b \ Q p : Z[] Z[]/ Z[]/ + p + 4, p 4 4 α 4ϖ + ϖ, α 3ϖ ϖ α, 3α 7ϖ + ϖ α, 4α 6ϖ ϖ α sl Q[] Q[] s slα s slα s sl α s sl α s 4 sl 4 α sl 4 α Q[], sl 4 α Q 4 + β 4ϖ 6ϖ + ϖ ϖ, β 3ϖ + ϖ β, 3β 7ϖ β sl Q[] Q[] s slβ s slβ s sl β s sl β s 4 sl 4 β ϖ β, 4β sl 4 β Q[], sl 4 β Q sl 4 α, sl 4 β slα 4 clα 3 ϖ + ϖ α 4β β 3α O ϖ ϖ α 3β β 4α ϖ L QE E L s 6 + s 6s 8 s 4 + { } a b GalL/Q, GL, F d c L Q 3 Abel Q 6 Abel

MeBo 4.9. 6:9 + 9 + slz s, clz c c s + s slz sc s c, clz c s + s c slz s clz c slz + z slzclz + clzslz slzslzclzclz sc s c c + c s + s c s sc s c s c s + s c c s + s c + c s s c s s c + s c s c s s + s s s + s + s + s s s s + s s s + s + s s s + s s + s s s + s + s s 4 + s s s 4 + s 4 s 4 + s s 4 s + s s s 4 ss4 + s + s + s + + s + s 4 + s + s + s + + s + ss4 + s 4 + slx + y slxsl y + sl xsly + sl xsl y sl z slz s, sl z s slz ss + s 4, s s 4 sl z ss sl 4 z + s 4 + s4 4 6s 4 s 4 4 + s4 4 6s 4 s 4 + s 4 + s 4 { + s4 + 4s s 4 }{ + s 4 4s s 4 } + s 4 s8 4s 6 + s 4 + 4s + s 8 + 4s 6 + s 4 4s + + s 4 s4 s s 4 + s + s 4 s4 s s 4 + s + s 4 sl z s s 4 s s s 4s 3 s s s 3 slz ss + s 4

3 s sl z ss + s 4 ss 4s 3 s + s 4 {s + ss } + s 4 8s 4 s + s 4 { s4 + s s 3 } + s 4 8s 4 s 4 + s 4 sl z { s4 + s s 3 } + s 4 4s 4 s 4 + s 4 3s4 + s 4 4s 4 s 4 + s 4 s8 6s 4 + + s 4 s4 + s s 4 s + s 4 3 slz slz ±, ± z m ϖ + n dr r 4 sl z z m 4 ϖ + n 4 ϖ m n mod s ± ±, ± ± r 4 + slz + z slzsl z + sl zslz + sl zsl z ϖ m n mod dr.6 ss + s 4 s + s4 + s s 4 s + s 4 s ss + + s 4 s + s4 ss + ss 4 + s s 4 s + s 4 4s 4 s 4 + s4 s s 4 + ss 4 + s s 4 s s 8 + s 4 + 4s 4 + 4s 8 s{s8 s 8 6s 4 + } s 8 s 4 + s{ s4 + }{s 4 } { s 4 + }{ + s 4 } s{s4 } { s 4 } s{ s8 + 6s 4 + } { s 4 + }{ + s 4 } slz z 3 s{s4 + } { + s 4 } L + β 4ϖ ϖ slβ 4 4 4

MeBo 4.9. 6:9 3 L α 4ϖ + ϖ slα 4 + 4 slβ slα 4 3 γ, γ + γ γ +, γ γ + sl + z ss4 γ γs 4, s s 4 γ sl z γs 4 slz sl + z sl + z sl 4 + z γ γsl 4 + z { ss 4 } [ { γ ss 4 γ } 4 γ] γs 4 γs 4 [ { ss 4 } 4 γ γ γs 4 ] ] ss 4 γ [s 4 s 4 γ 4 γγs 4 4 ] [γs 4 s 4 γ 4 γs 4 4 γs 4 ] ss 4 γ [s 4 s 4 γ 4 γγs 4 4 ] [γs 4 s 4 γ 4 γs 4 4 γs 4 s 4 x s 4 s 4 γ 4 γγs 4 4 xx γ 4 γγx 4 xx 4 4ax 3 + 6a x 4a 3 x + a 4 γa 4 x 4 4a 3 x 3 + 6a x 4ax + x 4ax 4 + 6a x 3 4a 3 x + a 4 x a 3 x 4 + a x 3 3ax + x + a x + a 3 4ax 4 + 6a x 3 + 4a 3 3ax + a 4 + x + a x + a + x 4 + a 3x 3 + 6a + 4x + a + x + a x + a x 4 + x 3 6x x + x γ x 4 + x 3 6x x + s 4 γ s 6 + s 6s 8 s 4 + s s γs 4 s 4 γ 4 γs 4 4 γs 4 s 6 s 6s 8 + s 4 +

3 slz slz sl 8 z sl 4 z + sl 6 z + sl z 6sl 8 z sl 4 z + sl 8 z sl 4 z + sl 6 z sl z 6sl 8 z + sl 4 z + 4 x 4 + x 3 6x x + s 6 + s 6s 8 s 4 + L Q 6 Abel Q 3 Abel s 4 x x 4 + x 3 6x x + Q 4 Kummer Q 8 4 4 S 4 4 8 4 3 Ferrar 3 4 3 y x + 6 x 4 + x 3 6x x + y 6 4 + y 6 3 6y 6 y 6 + y 4 4y + 84y 899 a + b + c + d ab + ac + ad + bc + bd + cd p 4 4 a, b, c, d abc + abd + acd + bcd q 84 abcd r 899 e ab + cd, f ac + bd, g ad + bc e, f, g 3 3 e + f + g ab + cd + ac + bd + ad + bc p 4, 4 3 ef + fg + ge ab + cdac + bd + ac + bdad + bc + ad + bcab + cd abca + b + c + abda + b + d + acda + c + d + bcdb + c + d 4abcd 899 4, 8 8, efg ab + cdac + bdad + bc abcda + b + c + d + a b c + a b d + a c d + b c d abcd{a + b + c + d ab + ac + ad + bc + bd + cd} + abc + abd + acd + bcd abcdab + ac + ad + bc + bd + cd r p + q rp q 4pr 84 4 4 899 43696 4

MeBo 4.9. 6:9 3 e, f, g 3 X 3 + 4 3X + 8 8 4x + 4 3 X 4 Y X 3 + 4 3X + 8 8 4X + 4 Y 3 + 6Y + 4Y + Y + Y + 4Y + Y 3 Y, 4 ± e ab + cd 3, f ac + bd 36 + 8, g ad + bc 36 8 a+b, c+d a+b+c+d, a+bc+d ac+ad+bc+bd f +g 7 a+b ± 7 ±, c+d 7 a+c ± 4 +, b + d 4 + a + d ± 4, b + c 4 4 y 4 + py + qy + r y ay by cy d y a a 4 + pa qa + r a a b a c a d qa aa + ba + ca + d a + ba + ca + d q 84 a + b, c + d, a + c 7 +, b + d 7 +, a + d 7, b + c 7 a + b + a + c b + c a 6 7 + 7 a + b + b + c a + c b 6 + 7 + + 7 a + c + b + c a + b c 6 7 + + 7 a + d + b + d a + b d 6 + 7 + 7 4 x 4 + x 3 6x x + x 4 + x 3 6x x + x + 3 + 6 + 7 + + 7 x + 3 + 6 7 + 7 x + 3 6 7 + + 7 x + 3 6 + 7 + 7 7 + 7 7 + + 7 7 34 4 9 s 4 3 + 6 + 9 Q ε + ϖ sl 4 3 + 6 + 7 + 7 4.7943474.933787

4 3 ϖ sl 4 3 + 6 + 7 3 4.7943474.933787 4ϖ sl 4 3 + 6 7 3 4.7338.477 Gauss sn.lemm 44 ϖ sl + ϖ 4 3 6 + 7 + 3 ϖ sl + ϖ 4 3 6 7 + 3 + + 4.3484.46867 +.46867 4.4996.9398 +.9398

ϖ MeBo 4.9. 6:9 6 sl ϖ 6 sl + β 4ϖ ϖ slβ 4 4 4 slβ α 4ϖ + ϖ slα 4 + 4 slα 4 8ϖ ϖ slα + β sl sl sl z sl z sl z 4 sl 4 z ϖ O + ϖ slα 4 +, slβ 4 sl α sl 4 α + ± sl β sl 4 β ± + sl α sl β 3 sl α +, sl β slα + β slαsl β + sl αslβ + sl αsl β 4 + + + 4 + 4 + 4 + 4 + + 4 + + + 4 + + 4 +

6 3 ϖ sl slα + β 4 + + + 4 + Excel 4 + + + 4 +.93378739379 7 ϖ sl 4 + + + 4 + 4 3 + 6 + 7 3 + a + b a, b + a b + ab a b, ab a ± +, b ± + ξ ε + ε + + + + + 4 4 6 + + + + + 3 + + 4 3 + 4 + 4 + 4 4 + 6 3 + 7 + 3 3 7 3 + 7 3 7 + 3 7 3 + 4 + 4 + 4 7 3 4 3 + 6 + 7 3

MeBo 4.9. 6:9 8 GalL/Q 7 8 GalL/Q L QE s 6 + s 6s 8 s 4 + { } a b GalL/Q, d c GL, F L Q 3 Abel K Q L K 6 Abel GalL/K Z[]/Z[] Z[]/ + Z[] Z[]/ Z[] F F L α 4ϖ + ϖ, + β 4ϖ ϖ ξ slα, η slβ ξ 4 +, η 4 L K ξ η L Q, ξ, η Galos, ξ, η 3 GalL/Q G G σ, τ, ρ σξ ξ, σ ση η, σ, τξ ξ, τ τη η, τ, ρξ ξ, ρ ρη η, ρ, G σ, τ, ρ σ 4 τ 4 ρ e, στ τσ, ρσ τρ, ρτ σρ σ, τ, ρ G {σ l τ m ρ n l,,, 3, m,,, 3, n, } G G p ndex p e {e} I I 7 L I {e, σ } Z/Z {e} J, J 6, J 9 P I {e, τ } Z/Z {e} J, J 4, J 9 P I 3 {e, σ τ } Z/Z {e} J 3, J, J 7, J 8, J 9, J, J P 3 I 4 {e, ρ} Z/Z {e} J P 4 I {e, σ 3 τρ} Z/Z {e} J P I 6 {e, σ τ ρ} Z/Z {e} J P 6 I 7 {e, στ 3 ρ} Z/Z {e} J P 7 J 4 {e, σ, σ, σ 3 } Z/4Z I F 6 N J 4 {e, τ, τ, τ 3 } Z/4Z I F 7 N J 3 4 {e, στ, σ τ, σ 3 τ 3 } Z/4Z I 3 F, F, F, F 9, F N 3 J 4 4 {e, σ τ, τ, σ τ 3 } Z/4Z I F 7 N 4 J 4 {e, σ 3 τ, σ τ, στ 3 } Z/4Z I 3 F, F 8, F N J 6 4 {e, στ, σ, σ 3 τ } Z/4Z I F 6 N 6 J 7 4 {e, σ ρ, σ τ, τ ρ} Z/4Z I 3 F 3, F 9, F N 7 J 8 4 {e, στρ, σ τ, σ 3 τ 3 ρ} Z/4Z I 3 F 4, F, F N 8 J 9 4 {e, σ τ, σ, τ } Z/Z I, I, I 3 F 3, F 4, F, F 6, F 7 N 9 J 4 {e, σ τ, σ 3 τρ, στ 3 ρ} Z/Z I 3, I, I 7 F 4, F 8, F 9 N J 4 {e, σ τ, ρ, σ τ ρ} Z/Z I 3, I 4, I 6 F 3, F 8, F N

8 3 F 8 F 8 F 3 8 F 4 8 F 8 F 6 8 F 7 8 F 8 8 F 9 8 F 8 F 8 H 6 H 6 H 3 6 {e, σρ, στ, σ τρ, σ τ, σ 3 τ ρ, σ 3 τ 3, τ 3 ρ} Z/8Z J 3 H M {e, τρ, στ, στ ρ, σ τ, σ τ 3 ρ, σ 3 τ 3, σ 3 ρ} Z/8Z J 3 H M {e, σ, τ, σ τ, ρ, σ ρ, τ ρ, σ τ ρ} D 8 J 7, J 9, J H M 3 {e, σ, τ, σ τ, στρ, στ 3 ρ, σ 3 τρ, σ 3 τ 3 ρ} D 8 J 8, J 9, J H M 4 {e, σ, τ, σ τ, στ, στ 3, σ 3 τ, σ 3 τ 3 } Z/4Z Z/Z J 3, J, J 9 H, H, H 3 M {e, σ, τ, σ τ, σ, σ 3, στ, σ 3 τ } Z/4Z Z/Z J, J 6, J 9 H 3 M 6 {e, σ, τ, σ τ, τ, τ 3, σ τ, σ τ 3 } Z/4Z Z/Z J, J 4, J 9 H 3 M 7 {e, στ 3, σ τ, σ 3 τ, ρ, στ 3 ρ, σ τ ρ, σ 3 τρ} D 8 J, J, J H M 8 {e, στ, σ τ, σ 3 τ 3, σ ρ, τ ρ, σ 3 τρ, στ 3 ρ} Z/4Z Z/Z J 3, J 7, J H M 9 {e, στ, σ τ, σ 3 τ 3, ρ, στρ, σ τ ρ, σ 3 τ 3 ρ} Z/4Z Z/Z J 3, J 8, J H M {e, στ 3, σ τ, σ 3 τ, στρ, σ ρ, τ ρ, σ 3 τ 3 ρ} Q 8 J, J 7, J 8 H M {e, σ, τ, σ τ, στ, στ 3, σ 3 τ, σ 3 τ 3, F 3, F 4, F, F 8, ρ, σ ρ, τ ρ, σ τ ρ, F 9, F, F G K στρ, στ 3 ρ, σ 3 τρ, σ 3 τ 3 ρ} {e, σ, τ, σ τ, στ, στ 3, σ 3 τ, σ 3 τ 3, τρ, τ 3 ρ, σρ, στ ρ, F, F, F G K σ τρ, σ τ 3 ρ, σ 3 ρ, σ 3 τ ρ} {e, σ, τ, σ τ, στ, στ 3, σ 3 τ, σ 3 τ 3, σ, σ 3, στ, σ 3 τ, Z/4Z Z/4Z F, F 6, F 7 G K 3 τ, τ 3, σ τ, σ τ 3 } I 3, J 3, J, J 9, F, F 8, F, H, H, H 3 I, I NI NI H 3, I 4, I, I 6, I 7 NI 4 NI NI 6 NI 7 F 8, J, J NI NI H 3, J 4, J 6 NJ 4 NJ 6 H 3, J 7, J 8 NJ 7 NJ 8 H, J, J NJ NJ H, F, F NF NF H, F 6, F 6 NF 6 NF 6 H 3, F 3, F 4 NF 3 NF 4 H, F 9, F NF 9 NF H I 4 P 4

MeBo 4.9. 6:9 8 GalL/Q 9 I 4 {e} I I I 3 I I 7 I 4 I 6 J J J 4 J 6 J 9 J 3 J J J J 8 J 7 F 7 F 6 F F F F 4 F 3 F 8 F 9 F F H 3 H H G Q Galos P 4 L P P P 3 P P 7 P 4 P 6 N N N 4 N 6 N 9 N 3 N N N N 8 N 7 M 7 M 6 M M M M 4 M 3 M 8 M 9 M M K 3 K K Q

3 3 L Q Galos K Q ξ 4 +, η 4 ε + Q L {e} Kξ, η P I P, P Kξ, η P I P, P Kξ, η P 3 I 3 Kξ, η, ξη P 4 I 4 P 4, P, P 6, P 7 Qslϖ/ P I P 4, P, P 6, P 7 P 6 I 6 P 4, P, P 6, P 7 Qslϖ/ P 7 I 7 P 4, P, P 6, P 7 N J N, N Kη N J N, N Kξ N 3 J 3 Kξη K 4 N 4 J 4 N 4, N 6 Kξη N J Kξη 3 N 6 J 6 N 4, N 6 Kξ η N 7 J 7 N 7, N 8 N 8 J 8 N 7, N 8 N 9 J 9 K ε Kξ, η N J N, N N J N, N M F M, M M F M, M M 3 F 3 M 3, M 4 Qξ + η Q ε M 4 F 4 M 3, M 4 Qξ η Q ε M F Kξ η K M 6 F 6 M 6, M 7 Kη K M 7 F 7 M 6, M 7 Kξ K + M 8 F 8 Qξ 3 η + ξη 3 Q + / M 9 F 9 M 9, M Q 4 M F M 9, M Q 4 M F K H Q K H Q K 3 H 3 K Q ξ + ε /ε, η + ε + /ε ξ η ε ξ + η /ε σ ε /ε, τ ε /ε

MeBo 4.9. 6:9 8 GalL/Q 3 L Galos ϖ sl 4 3 + 6 + 7 3 4 + + + 4 + ξ + + η + ξ η P 4 Galos ρ P 4 N J σ τ ξ + + η ξ + + η + ξ η + ξ η Q 6 4 ϖ sl 3 + 6 + 7 3 M 8 Qξ 3 η + ξη 3 Q + / P 4 /M 8 Galos 4 P 4 /M 8 L/N 4 Kummer ε + ξ ε ε, η ε + ε, ε ε + ε ξ + η Q 4 Q σ ε ξ + η ε τ ε ξ η ε ρ ε ξ + η + ε Q ε Q 4 Galos F 3 {e, σ, τ, σ τ, ρ, σ ρ, τ ρ, σ τ ρ} G F 3 NF 3 H F 3 στf 3 F 3 F 4 σf 3 σ τf 3 τ {e, σ, τ, σ τ, στρ, στ 3 ρ, σ 3 τρ, σ 3 τ 3 ρ} Q σ ε Q Q Q ε ε Q ε Galos Q ε, ε Q ε, J 9 F 3 F 4 {e, σ, τ, σ τ } G G/J 9 σ, τ, ρ σ τ ρ e, σ τ τ σ, ρ σ τ ρ, ρ τ σ ρ D 8 + J 9 Q, J 9 Q ε, F 7 F 6 F F 4 F 3 Q + Q, Q Q ε Q ε H 3 H H G Q Q Q Q

3 3 + 3 α ξ 3 η + ξη 3 Qα M 8 F 8, GalM 8 /Q Z/4Z Q 4 α α + α, α 4 α + M 8 x 4 x + + + x 4 x + α, β, γ, δ 4 αγ α γ α α + α 4 α α 3 3α 4 α α x 4 x + Qα Q 4 Galos Abel Galos σ : Qα Qα σα γ α 3 3α σγ α 3 3α 3 3α 3 3α α 9 9α 7 + 7α 3α 3 + 9α α 4 α α 4α 3 + α α α β GalQα/Q σ 4 Qα Q α Qα, Qα Q Q GalQα, /Q Z/4Z Z/Z < σ, ρ > ρ