Winter School on Galois Theory, Volume 1

Published by the

UNIVERSITY OF LUXEMBOURG

ISBN: 978-2-919940-08-0

ISSN: 1024-1833

UNIVERSITY OF LUXEMBOURG Campus Kirchberg

Mathematics Research Unit

Travaux mathématiques

Presentation

The journal Travaux mathématiques is published by the Mathematics Research Unit of the University of Luxembourg. Even though the main focus of the journal is on original research articles, surveys and historical studies are also welcome.

Editors-in-Chief

Carine Molitor-Braun (University of Luxembourg) Norbert Poncin (University of Luxembourg) Address:

University of Luxembourg Mathematics Research Unit 6, rue Coudenhove-Kalergi L-1359 Luxembourg

Grand-Duchy of Luxembourg

Email: carine.molitor@uni.lu or norbert.poncin@uni.lu poncin@cu.lu

Associate Editors

Jean Ludwig (University Paul Verlaine, Metz – University of Luxembourg) Martin Olbrich (University of Luxembourg)

Martin Schlichenmaier (University of Luxembourg) Anton Thalmaier (University of Luxembourg) Editorial Board

Bachir Bekka (University of Rennes)

Lionel Bérard-Bergery (University H. Poincaré, Nancy) Martin Bordemann (University of Haute Alsace, Mulhouse) Johannes Huebschmann (University of Lille)

Pierre Lecomte (University of Liège) Jiang-Hua Lu (University of Hong Kong)

Raymond Mortini (University Paul Verlaine, Metz) Jean-Paul Pier (University of Luxembourg) Claude Roger (University Claude Bernard, Lyon) Izu Vaisman (University of Haifa)

Alain Valette (University of Neuchâtel)

Special issue

2

Proceedings of the Winter School on Galois Theory (Volume I) Universit´e du Luxembourg, 15 – 24 February 2012

Luxembourg Organisers Sara Arias-de-Reyna Lior Bary-Soroker Gabor Wiese Sponsors

Fonds National de la Recherche Luxembourg, Luxembourg Foundation Compositio Mathematica, The Netherlands DFG-Sonderforschungsbereich TRR 45, Germany

Preface

The work of Evariste Galois (1811 – 1832), whose 200th birthday was recently celebrated, is at the origin of a theory that plays a key role in many mathematical disciplines, such as number theory, algebra, topology, and geometry.

From 15 until 24 February 2012 we organised a Winter School on Galois Theory at the University of Luxembourg, which attracted almost 100 participants. In the winter school two very important topics were united: on the one hand, in recent years there have been tremendous developments in number theory, such as the proofs of two famous conjectures: Serre’s Modularity Conjecture and the Sato-Tate Conjecture. It is becoming clear that these are only some instances of a bigger picture encompassed in new directions in the Langlands programme. Galois representationsare at the core of all of this.

On the other hand, in Galois field theory, motivated by the famous inverse Galois problem first considered by Hilbert, the so-called regular inverse Galois problem could be solved over various classes of fields. This has been done by developing “patching”, an algebraic analog of the “cut-and-paste” constructions in complex geometry. The method of patching has been further developed and ex-panded and achieved results both in Galois theory and algebra, e.g., in differential Galois theory, division algebras and quadratic forms.

The communities of people working on Galois representations and those work-ing in Galois field theory currently only seem to be insufficiently interactwork-ing, although for a large part they are studying the same kind of objects, namely absolute Galois groups. One instance, where the possible connection was success-fully exploited, is in the proof of potential modularity: a classical result of field arithmetic on the field of all totally real numbers is essential.

Both groups use their own very advanced sets of tools. On the one hand, at the winter school the statement of Serre’s Modularity Conjecture, some of the tools involved in the proof of it, as well as some generalisations and connected open problems were presented. On the other hand, several of the patching techniques and various applications were explained.

The winter school consisted of three ‘preparatory days’, whose content was accessible to Master students, whereas the main week of the school addressed mainly PhD students or young postdocs. The current Volume I unites two sets of lecture notes, which are very elaborate versions of two courses given during the preparatory days.

counterexam-4

ples that show why certain hypotheses are necessary. One also finds a chapter on the history of field theory as well as other historical remarks throughout the text. The second set of notes addresses the theory of profinite groups. They are written by Luis Ribes, who is the author of two standard books on this subject. Being necessarily much shorter than the two books, these notes have the feature of presenting an overview stressing the main concepts and the links with Galois theory. Since for those proofs which are not included in the notes precise references are given, we think that the notes, due to their conciseness and nevertheless great amount of material, constitute an excellent starting point for any Master or PhD student willing to learn this subject.

We wish to express our sincere gratitude to all speakers at the winter school for their excellent lectures, and for having taken a lot of time and care in elaborating their notes for these proceedings.

May the readers enjoy them and may they serve well! The Editors

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,2 36 &(<#%.&/, #(#7#6$ a *>#. K /0 ,&((#) )#+ % $!#8 /+ f = MinPol(a|K) /0 0#4&.&%(#8 *$-#.:/0# &,)#+ % $!#1 ?-# #(#7#6$ a /0 ,&((#) +*%#!2 &,)#+ % $!# *>#. K8 /+ degsepf = 18 /1#1 /+ f -&0 *6(5 *6# .**$ /6 eK1 B#7&.C $-&$ $-#

#(#7#6$0 /6 K &.# 0#4&.&%(# &6) 49.#(5 /60#4&.&%(# *>#. K1 ;+ a /∈ K /0 49.#(5 /60#4&.&%(# *>#. K $-#6 p > 0 &6) ape

∈ K :/$- e = expinsMinPol(a|K)1

)2 36 &(<#%.&/, #@$#60/*6 L|K /0 ,&((#) )#+ % $!#8 /+ &(( a ∈ L &.# 0#4&.&%(# *>#. K8 *$-#.:/0# &,)#+ % $!#1 ;+ S ⊆ eK /0 & 0#$ *+ &(<#%.&/, #(#7#6$0 :-/,- &.# 0#4&.&%(# *>#. K8 $-#6 K(S)|K /0 & 0#4&.&%(# '#() #@$#60/*61

3 '6/$# #@$#60/*6 L|K /0 /60#4&.&%(# /= $-# (/6#&. $.&,# +*.7 TrL|K *6 $-#

KD04&,# L >&6/0-#01 ;+ L|K /0 0#4&.&%(# *+ '6/$# )#<.## [L : K] = n8 $-#6 n #(#7#6$0 b1, . . . , bn/6 L &.# & KD%&0/0 *+ L /= $-#/. -&)'%&.&, ,/

∆(b1, . . . , bn) = det(TrL|K(bi· bj))1≤i,j≤n

"#$% &#' ()&*%+, -. L = K(α) )&" f = MinPol(α|K) '+$& ∆(1, α, . . . , αn−1_{) =}

∆(f ),

$/ 0+$ )12$34)*5 $6'$&%*#& L|K *% 5)11$" !"#$% &'(# )")*$#7 *. )11 a ∈ L )4$

894$1: *&%$8)4)31$, -. S ⊆ eK *% ) %$' #. )12$34)*5 $1$;$&'% <+*5+ )4$ 894$1:

*&%$8)4)31$ #($4 K7 '+$& K(S)|K *% ) 894$1: *&%$8)4)31$ =$1" $6'$&%*#&, -. *& 5)%$ p > 0 '+$4$ *% )& $68#&$&' e <*'+ Lpe

⊆ K7 <$ 5)11 '+$ %;)11$%' %95+ e '# 3$ '+$ #+ ,'#'- ,. &'(# )")*&$&-% expins(L|K) #. L|K7 #'+$4<*%$ *' *%

=∞, -. L = K(a) *% 894$1: *&%$8)4)31$ #($4 K <*'+ f = MinPol(a|K) '+$&

expins(L|K) = expinsf,

! "! #$%&%'()(%*+ *"+ K ," - ."#$/ char K = p ≥ 00 )/ 12 +&" -#3",(-!4 "5+"67!'67 M|L -6$ L|K -(" 7"8-(-,#"/ 7' !7 +&" "5+"67!'6 M_|K0 3/ 12 p > 0 -6$ a ∈ KrKp/ +&"6 Xpe− a !7 -6 !67"8-(-,#" !(("$94!,#" 8'#): 6';!-# !6 K[x] '2 $"3("" pe _{2'( -6) e ∈ N0} 5/ 12 p = 0 +&"6 -## -#3",(-!4 "5+"67!'67 '2 K -(" 7"8-(-,#"0 "/ 12 p > 0 -6$ K = Kp _{+&"6 -## -#3",(-!4 "5+"67!'67 '2 K -(" 7"8-(-,#"0}

$/ 0+$ =$1"% %)'*%.:*&2 5/ #4 "/7 *,$, =$1"% +)(*&2 #&1: %$8)4)31$ )12$34)*5 $6'$&%*#&%7 )4$ 5)11$" #".#/-7 '+$ #'+$4 =$1"% &0 #".#/- ! ₀

!"#$%&'

! Fp "# $%&'%()* )+% &,)"-.,/ '0.()"-. 1%/2 Fp(x).-)!

3! 45%&6 1%/2 K 7")+ char K = p > 0 "# (-.),".%2 ". , #8,//%#) $%&'%() 1%/2* )+% !"#!$% $&'()"! -' K* .,8%/6 )+% 0."-. Kp−∞ -' )+% ,#(%.2".9 #%:0%.(% -' 1%/2# K ⊆ K1/p _{⊆ K}1/p2 ⊆ . . . ⊆ K1/pe ⊆ . . . . ./ <="() -#3",(-!4 "5+"67!'6 '2 - 8"(2"4+ ."#$ !7 8"(2"4+0

! ! #$%&%'()(%*+ *"+ K ," - ."#$ >!+& char K = p ≥ 0/ -6$ L|K ," -6 -#3",(-!4 "5+"67!'60

)/ %&"(" !7 - ;-5!;-# 79,."#$ Lsep '2 L|K 794& +&-+ Lsep|K !7 7"8-(-,#"/

6-;"#)

Lsep={a ∈ L ; a !7 7"8-(-,#" '="( K} .

! !"#$%&'(') *'+') ,-'. L|Lsep &/ 0!)'"+ &./'01)12"'3 ' 4'.5(' (-' !"#!! $%

&!'(#()*+*,-)'/03 (-' !"#!! $% *.&!'(#()*+*,- 5# (-' '6('./&5. L|K 2+

[L : K]sep= [Lsep: K] )'/03 [L : K]ins= [L : Lsep] .

,-' "1((') &/ 1"71+/ 1 057') 5# p 8)'/03 9 &# p = 0:3 ;# L = K(a) &/ 1 /&<0"' 1"='2)1&> '6('./&5. 1.4 f = MinPol(a|K) (-'.

[L : K]sep= degsepf , [L : K]ins= deginsf .

"# !"#$"% #& '!" ($)%)')*" "+"%",'- ?'( L|K 2' 1 @.&(' '6('./&5. 5# @'"4/3 ,-'. (-' #5""57&.= /(1('<'.(/ 1)' 'A!&B1"'.(C !

$ # ,-' '6('./&5. L|K &/ /&<0"'D &3'3 -1/ 1 0)&<&(&B' '"'<'.(3 $%# ,-' '6('./&5. L|K -1/ 5."+ @.&('"+ <1.+ /!2@'"4/3

$&# ' -1B' p = 0 5) p > 0 1.4 [L : K]ins= pexpins(L|K) 3

$!# dimLΩL|K ≤ 1 $'() *' +),*'*)-. /01 1 2#1

/# .#$#++/$0- EB')+ /'01)12"' @.&(' '6('./&5. &/ /&<0"'3 F"" @.&(' '6('.$ /&5./ 5# (-' @'"4 K 1)' /&<0"' &G K &/ 0')#'>( 5) [K : Kp_{] = p3}

!"#$%& !" K = Fp(x, y)#! "$! %&"'()&* +,)-"'() .!*/ ') "0( 1&%'&#*!2 (1!%

Fp3 4$!) K|Kp'2 )(" 2'56*!7 &)/ &) ').)'"! +&5'*8 (+ /'9!%!)" 2,#.!*/2 '2 :'1!)

#8 {Kp_{(y + x}n_)} n∈Z, p∤n3 30),4 )5*6 /4765 /(846, *' 79:,"47*/ ;,9< ,=),'6*('6 9,) 86 /('6*<,4 )47'6/,'<,')79 ;,9< ,=),'6*('61 12132 4"&),)')#,- >,) L|K ", 7 ;,9< ,=),'6*('1 3 68"6,) A ⊆ L *6 /799,< K/(+"!)#(*0(++- *. !'!. !., *0 '( ;'*), 68"6,) {a1, . . . , an} (0 A 67)*6;,6 7 ?(9@'(A*79 4,97)*('. *1,1 *0 f ∈ K[x1, . . . , xn] , f (a1, . . . , an) = 0 =⇒ f = 0 (aν ∈ A <*6)*'/)) ! _{!""#$ %&'⇒%('$ )# L = K(α) *+, p > 0 -./+ Lsep= K(β)01-. β = α}pe 2 3" [L : Lsep] = pe

*+, e = expins(L|K)4 %('⇒%5'$ )# [L : Lsep] = pe*+, e = expins(L|K) -./+ L = Lsep(α)

"#$%&' K()$*+,-)./)$$0 .1%+2+1%+13 +$+4+13& x15 ' ' ' 5 xn )-+ )$&# /)$$+% !"#$ %#&' !(%#) #- *(& (+,#) #6+- 3"+ 7+$% K' !"#$%&' !"# L|K $" % &"'( ")#"*+,-* %*( a1, . . . , an∈ L $" %'."$/%,0%''1 ,*("2 3"*("*# -4"/ K5 %6 78 A = (αij)∈ GLn(K)#9"* #9" "'":"*#+ b1=Pjα1jaj; 5 5 5 ; bn=Pjαnjaj %/" %'."$/%,0%''1 ,*("3"*("*# -4"/ K5 $6 <-/" ."*"/%''1; '"# f1, . . . , fn∈ K[x1, . . . , xn]$" n 3-'1*-:,%'+ =,#9 *-* 4%*2 ,+9,*. 8>*0#,-*%' ("#"/:,*%*# ∂(f1, . . . , fn) ∂(x1, . . . , xn)= det  ∂fi ∂xj  i,j=1...n = det A ,* 0%+" %6 #9"* b1= f1(a1, . . . , an), . . . , bn= fn(a1, . . . , an)%/" %'."$/%,0%''1 ,*("3"*("*# -4"/ K5 ! ! "! #$%&%'()(%*+ *"+ L|K ," - ."#$ "/+"01!'02 )8 %&"(" -(" 3-/!3-# K4-#5",(-!6-##) !0$"7"0$"0+ 1"+1 A !0 L -0$ -## &-8" +&" 1-3" 6-($!0-#!+)2

,8 9".& /#44#1 /)-%.1)$.30 .& /)$$+% 3"+ %&(!)-#!"#!-# "#.&## trdeg(L|K) #: L|K )1% +6+-0 &;/" 4)<.4)$ A .& /)$$+% ) %&(!)-#!"#!-# +()# #: L|K2 '( -0) 196& A +&" "/+"01!'0 L|K(A) !1 -#5",(-!62 =: L = K(A) :#- #1+ &;/" A5 3"+ +<3+1&.#1 L|K .& /)$$+% /0&#,1 %&(!)-#!"#!%(,2

/8 :; L = K(S) - +(-016"0$"06" ,-1" '; L|K 6-0 ," 6&''1"0 -1 19,1"+ '; S2 %8 :; L1 ⊆ L2 -(" "/+"01!'01 '; K +&"0 trdeg(L1|K) ≤ trdeg(L2|K)2 :;

trdeg(L1|K) !1 .0!+" +&"0 +&" !0"<9-#!+) ,"6'3"1 -0 "<9-#!+) != L2|L1 !1

-#5",(-!62 +8 %&" ."#$ "/+"01!'0 L|K !1 .0!+"#) 5"0"(-+"$ != -0) 1+(!6+#) !06("-1!05 1"4 <9"06" '; 19,."#$1 !1 .0!+"2 >0) 19,"/+"01!'0 '; - .0!+"#) 5"0"(-+"$ ."#$ "/+"01!'0 !1 -5-!0 .0!+"#) 5"0"(-+"$2"! :8 >0 -#5",(-!6-##) 6#'1"$ ."#$ !1 $"+"(3!0"$ 97 +' !1'3'(7&!13 ,) +&" +(-04 16"0$"06" $"5("" '8"( +&" 7(!3" ."#$ -0$ !+1 6&-(-6+"(!1+!62 ! _{! "#$ %&!'(&'$ )* +,+- "#./ *)%%)0/ *1)2 "#$ $3(&".)!/ dbν =}Pn i=1∂f∂xνi(a1, . . . , an) dai .! Ω_K(a1,...,an)|K,

"! _{4)1$ 51$6./$%78 * L|K ./ '$!$1&"$9 :7 n $%$2$!"/ "#$! &!7 /(:$;"$!/.)! L◦|K ./ '$!$1&"$9}

:7 ≤ n + 1 $%$2$!"/< 6*, +,+-,',=, * trdeg(L|K) ≤ 1 "#$! n /(>6$< :(" !)" .! '$!$1&%8 * G = Z/8Z)5$1&"$/ )! L = Q(x1, . . . , x8) :7 676%.6&%%7 5$12(".!' "#$ ?&1.&:%$/ xi "#$! L ./

@&%)./ 0."# '1)(5 G )?$1 "#$ A;$9 A$%9 L′_{= Fix}

GL:7 B,C, D(" L′./ !)" 5(1$%7 "1&!/6$!9$!"&%

! !"#$%&'(') *'+') "## $%&$'()* +&, -'*.#)* .( )% &%/ +-' +) #'+*) 00 1'+-* %#, +&, $%&)+2&', 2& + (+('- %3 4)'2&2)5 3-%6 7 0 8$39 :9;<9 =') .* &%/ $%6' )% *%6' &'/'- $%&$'()*> ,'?'#%(', 2& )@' A-*) $@+()'- %3 )@' !"#$%&'!#( %3 "9 B'2# 8 7C!<9 D%- (-%%3* *'' +#*% E=+&F 77GH %- EDIH9

! "! #$%&'&(&)'* =') F J' + A'#, /2)@ *.JA'#,* L +&, M /@2$@ $%&)+2& + $%66%& *.JA'#, K9

+< L +&, M +-' $+##', !"#$% & '!()*!"+ *,#% K 23 )@' $+&%&2$+# 6+( L⊗KM → F >

F2?'& J1 x ⊗ y 7→ x · y> 2* 2&K'$)2?'> 29'9 23 KL#2&'+-#1 2&,'('&,'&) '#'6'&)* %3 L +-' ML#2&'+-#1 2&,'('&,'&) 8+&, )@'& KL#2&'+-#1 2&,'('&,'&) '#'6'&)* %3 M +-' LL#2&'+-#1 2&,'('&,'&)<9 B' ,'&%)' )@2* J1 + -'$)+&F.#+- ,2+F-+6M

L LM

K M

N3 [L : K] < ∞> )@2* 2* 'O.2?+#'&) )% *+1 )@+) [L : K] = [LM : M]9 N3 L %- M 2* +#F'J-+2$ %?'- K> )@' -*./*(!+0. LM %3 )@' )/% A'#,* L +&, M 2& F 2*

'P+$)#1 )@' 26+F' %3 L ⊗KM 2& F > %)@'-/2*' )@' 26+F' 2* + *.J,%6+2& /2)@

O.%)2'&) A'#, LM9

Q%-' F'&'-+##1 *.JA'#,* L1, . . . , Ln %3 F |K +-' $+##', !"#$% & !"'#/#"'#"+ 23

)@' 6.#)2(#2$+)2%& L1⊗K. . .⊗KLn→ F 2* 2&K'$)2?'9

J< L +&, M +-' $+##', $ 1#2%$!-$ & !"'#/#"'#"+ ! _{*,#% K 23 KL+#F'J-+2$+##1}

2&,'('&,'&) '#'6'&)* %3 L +-' ML+#F'J-+2$+##1 2&,'('&,'&) 8+&, )@'& KL+#L F'J-+2$+##1 2&,'('&,'&) '#'6'&)* %3 M +-' LL+#F'J-+2$+##1 2&,'('&,'&)<9 N3

trdeg(L_{|K) < ∞ )@2* 2* 'O.2?+#'&) )% trdeg(L|K) = trdeg(LM|M)9}

$< 8Q+$=+&' 7G7< R@' 'P)'&*2%& L|K 2* $+##', (#/$%$2 # 23 char K = 0 %- 23 char K = p > 0 +&, L 2* #2&'+-#1 ,2*K%2&) 3-%6 )@' ('-3'$) $#%*.-' Kp−∞

|K

%3 K9 R@2* 2* 'O.2?+#'&) )% *+1 )@+) L 2* #2&'+-#1 ,2*K%2&) 3-%6 K1/p _{%?'- K9}

D%- +#F'J-+2$ 'P)'&*2%&* )@2* ,'A&2)2%& %3 *'(+-+J#' $%2&$2,'* /2)@ )@' 3%-6'-,'A&2)2%& 2& 979,9

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J-+2$ $#%*.-' eK|K %3 K9

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! _{!"# $%&'( !""}

! "# f ∈ K[x1, . . . , xn]$% &' $(()*+,$-.) /0.1'02$&. 34)' 34) #+',3$0' 5).* 0# 34)

41/)(%+(#&,) f = 0 $% ()6+.&( 07)( K $8 f $% !"#$%&'$( )**'+%,)!$'9 $!)! $(()*+,$-.) 07)( eK!

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>? $'3)6(&.@ 07)( eK! A) ,&.. %+,4 %,4)2)% - *)'&)'" 07)( K!

! "! #$%&%'()(%*+ *"+ F ," - ."#$ /!+& 01,."#$0 K, L, M, N /!+& !23#10!'20

K_{⊆ L} -2$ K_{⊆ M ⊆ N .} "# ,%-.$ &$%&.$)/+ L -2$ N -(" #!2"-(#) $!04'!2+ '5"( K !6 L -2$ M -(" #!2"-(#) $!04'!2+ '5"( K7 -2$ LM -2$ N -(" #!2"-(#) $!04'!2+ '5"( M8 L LM LN K M N $# 9: L -2$ M -(" #!2"-(#) $!04'!2+ '5"( K +&"2 L ∩ M = K; %&'()(*+)( +,( -+./0 0)+1 2'( 2()3 !"# K 4, 2'( ,+24+, +* /4,(") 0456+4,2,(55#7 !"#$%& B)3 α, β -) *$8)()'3 ,02/.)= (003% 0# 34) /0.1'02$&. x3_{+ 2! <4)'} Q(α)∩ Q(β) = Q9 -+3 34) 5).*% &() '03 .$')&(.1 *$%C0$'3 07)( Q! -# 9: L|K !0 <-#'!0 =3:; 87 > -2$ L∩M = K7 +&"2 L -2$ M -(" #!2"-(#) $!04'!2+ '5"( K; 0# 9: L, M -(" -#?",(-!3-##) !2$"@"2$"2+ '5"( K -2$ '2" ': +&"A !0 ("?1#-( '5"( K +&"2 L, M -(" #!2"-(#) $!04'!2+ '5"( K; ! 0! #$%&%'()(%*+ *"+ L|K -2$ M|K ," 01,."#$0 ': F |K; "# *"+ L ⊆ M; 9: L|K -2$ M|L -(" 0"@-(-,#"7 ("0@; ("?1#-(7 "B+"20!'207 0' !0 M_{|K; C'25"(0"#) !: M|K !0 0"@-(-,#"7 ("0@; ("?1#-(7 0' !0 L|K;}

!"#$%& D+3 M|L '))* '03 -) %)/&(&-.) ()%/! ()6+.&(E B)3 char K = p > 09 M = K(x)&'* L = K(xp_{)! <4)' M|K $% ()6+.&( -+3 M|L $% '03 )7)' %)/&(&-.)!}

"# char K = 0 34) %&2) )=&2/.) #0( &'1 $'3)6)( p > 1 %40F% 34&3 M|L $% '03 ()6+.&(!

$# %&" "B+"20!'2 L|K !0 0"@-(-,#" !6 "5"() .2!+"#) ?"2"(-+"$ 01,."#$ L◦ &-0

- +(-203"2$"23" ,-0" (a1, . . . , an)/!+& L◦|K(a1, . . . , an)0"@-(-,#" -#?",(-!3

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$ 5# L|K &6 6'3-)-0"' (7'. LM|M (884

%$ 5# L|K -./ M|K -)' 6'3-)-0"' (7'. -"68 (7' 289386&(!9 LM|K4

!"#$%& !" K = Fp(u)#$"% &' $'()")*+$'&") u, -!" L = K(x, y) #$"%

yp_{= x&'( M = K(x, z) #$"% z}p_{= x + u. /%)' L|K &'( M|K &*) 0)-&*&12)}

1!" LM|K '3".

&# 5# K &6 -"1'0)-&2-""+ 2"86'/ &. L -./ M|K &6 - 6&93"' -"1'0)-&2 ':('.6&8. (7'. L -./ M -)' "&.'-)"+ /&6;8&.( 8<') K4

'()*"+,--./ 5# K &6 3')#'2( 8) [K : Kp_{] = p = char K= -./ &# K &6 -"1'$}

0)-&2-""+ 2"86'/ &. L (7'. L|K &6 )'1!"-)4

!"#$%& 4)" K = Fp(a, b)1) "%) 5)2( 36 *&"$3'&2 6!'7"$3'0 $' "#3 8&*$&12)0 a, b

38)* Fp. /%) )9!&"$3' yp= x2p+ axp+ b$0 $**)(!7$12) 38)* K, 03 ()5')0 & -!*)2:

$'0)-&*&12) 5)2( );")'0$3' L = K(x, y)|K(x) 36 ()<*)) p. K $0 &2<)1*&$7&22: 7230)( $' L, 1!" L &'( K1/p _{&*) '3" 2$')&*2: ($0=3$'" 38)* K 0$'7) [K}1/p_{: K] = p}2 _1!"

[LK1/p_{: L] = p1)7&!0) 36 b}1/p_{= y− x}2_{− a}1/p_x.

*# ,'( L, M 0' -"1'0)-&2-""+ &./'3'./'.( ':('.6&8.6 8# K4

$ 5# L|K &6 )'1!"-) (7'. L, M -)' "&.'-)"+ /&6;8&.( 8<') K -./ LM|M &6 )'1!"-) (884

%$ 5# L|K -./ M|K -)' )'1!"-)= (7'. -"68 (7'&) 289386&(!9 LM|K4

!"#$%& L = Q(x, y) #$"% y2 _{= x &'( M = Q(x, z) #$"% z}2 _{= 2x &*)}

*)<!2&* );")'0$3'0 36 Q, 1!" "%)$* 73+-30$"!+ LM = Q √2, y
'3".

0# >7' ':('.6&8. L|K &6 )'1!"-) &? L|K &6 6'3-)-0"' -./ K &6 -"1'0)-&2-""+ 2"86'/ &. L4 >/%) &00!+-"$3' !"#$#%&! 7&''3" 1) (*3--)(? 0)) );&+-2) "3 (.@

1# !"#$%&'() *&'(!&'+,- L|K &6 )'1!"-) &# (7')' &6 - <-"!-(&8. v @2#4 A4B4CD 8# L|K E&(7 )'6&/!' F'"/ κ(v) = K4 23"4 5,-3,6+78( 97: )-,"*(# ,:* ",--*& K !"#$%&"'$

A!" '3" )8)*: *)<!2&* );")'0$3' L|K %&0 & KB*&"$3'&2 -2&7)C L = R(x, y) #$"% x2_{+ y}2_{+ 1 = 0$0 *)<!2&* 38)* K = R, 1!" &22 '3' "*$8$&2 8&2!&"$3'0 36 L|K %&8)}

*)0$(!) 5)2( C.

! "! #$%&'(%()*+ $) ,%&(-.(/* .,('0+ "#$%&'()**$+ !,-.

/$0 K 1$ 2 3$*4 5%0' char K = p > 06 *$0 L|K 1$ 27 $80$79%:7 :; K< =:+ 2

>:*?7:(%2* f ∈ L[X1, . . . , Xn] 5$ &2** ∂f = max{degXνf ; 1 ≤ ν ≤ n} 0'$

!"#$!% &'("'' :; f<

2. 1(.,),*,$)2 @ 9A19$0 X ⊂ L %9 &2**$4 p)$*&' '*&'*#6 %; ;:+ 27? 37%0$ 9$0 x1, . . . , xn:; 4%B$+$70 $*$($709 %7 X f _{∈ L}p_[X 1, . . . , Xn], ∂f < p, f (x1, . . . , xn) = 0 =⇒ f = 0 (∗) ':*496 %<$< %; 0'$ >:5$+ >+:4A&09 xe1 1 · · · xenn 5%0' 0 ≤ eν < p 2+$ LpC*%7$2+*? %74$>$74$70< 1. 1(.,),*,$)2 D28%(2* pC%74$>$74$70 9A19$09 :; L 2+$ &2**$4 p)+!,', :; L< %&") *(" '+ ,&" -*." /*($!0*#!,) 5'%&' %9 &2**$4 0'$ $- '".'/# '0 1*'*# iexp(L) :; L< E; %0 %9 37%0$ 5$ '2F$ piexp(L)_{= [L : L}p_]<

1 -23-", X ⊂ L !- * p43*-" !5 !, !- * .!0!.*# -", '+ 6"0"(*,'(- +'( L|Lp_<

&. !"#$%&'

!" iexp(L) = 0 ⇐⇒ L #$ %&'(&)*"

+" [L : K] < ∞ =⇒ iexp(K) = iexp(L)"

," -( L|K #$ $&%.'./0& .01&/'.#) *2&3 . p4/.$& 5( K #$ . p4/.$& 5( L6 $5 iexp(K) = iexp(L)" 7$%&)#.008 9& 2.:& iexp(L) = iexp(Lsep_{) 92&'& L}sep _{#$ *2& $&%.'./0&}

.01&/'.#) )05$;'& 5( L6 )(" ,"<"

=" -( L = K(x) #$ . '.*#53.0 (;3)*#53 >&0? *2&3 iexp(L) = 1 + iexp(K)"

@" -( L #$ >3#*&08 1&3&'.*&? *2&3 iexp(L) = trdeg(L|Fp)6 A5'& %'&)#$&08 B p4/.$& 5(

L#$ . *'.3$)&3?&3)& /.$& 5( L C/;* 35* )53:&'$&08D"

E" F5'& 1&3&'.008 -( L|K #$ >3#*&08 1&3&'.*&? 5( *'.3$)&3?&3)& ?&1'&& n *2&3 iexp(L) = n + iexp(K)"

4. 1(.,),*,$)2 @ 9A19$0 X ⊂ L %9 &2**$4 p)$*&' '*&'*# 12'" K %; &:74%0%:7 (∗)

%9 920%93$4 ;:+ 2** >:*?7:(%2*9 f ∈ Lp_K[X

1, . . . , Xn] 5%0' ∂f < p< @ p)+!,' :;

L|K %9 2 (%7%(2* 9$0 :; G$7$+20:+9 ;:+ L|KLp_{:+ $HA%F2*$70*? 2 (28%(2* 9A19$0}

%7 L :; $*$($709 pC%74$>$74$70 :F$+ K< %&" /*($!0*#!,) '+ *## p43*-"- '+ L|K !- ,&" -*." 274 %9 &2**$4 0'$ "'%!#$2' $- '".'/# '0 1*'*# iexp(L|K) :; L|K< E7 0'$ 37%0$ &29$ 5$ '2F$

! !"#$%&'(') *'+') "# !"#$%&'

!" #$ K ⊆ Lp_{%&'( iexp(L|K) = iexp(L)) *+,' -,'./0'12 3 p4560' +$ L|K /0 6 p4560'}

+$ L 6(7 .+(8',0'12"

9" #$ L′_{|L 6(7 K}′_{|K 6,' 0'-6,651' 61:'5,6/. ';%'(0/+(0 6(7 /$ K}′_{⊆ L}′_%&'(

iexp(L′|K′) = iexp(L|K).

<0-'./6112 =' &68' iexp(L|K) = iexp(Lsep_|Ksep_{)) 6(7 iexp(L|K) = 0 /$ L|K /0}

0'-6,651' 61:'5,6/." >" ?'% L|K 5' @(/%'12 :'(',6%'7" A&'( iexp(L|K) = 0) /"'" Lp_{K = L) /B L|K /0} 0'-6,651' 61:'5,6/." C" #$ L|K /0 @(/%'12 :'(',6%'7 6(7 0'-6,651' %&'( iexp(L|K) = trdeg(L|K). D" #$ L|K /0 @(/%'12 :'(',6%'7 %&'(

trdeg(L|K) ≤ iexp(L|K) ≤ iexp(K) + trdeg(L|K) .

<EF61/%2 +( %&' 1'$% 6--'6,0 /B L|K /0 0'-6,651'" <EF61/%2 +( %&' ,/:&% 6--'6,0 '":" $+, %&' ,6%/+(61 $F(.%/+( @'17 L = K1/p_(x

1, . . . , xn)/( n /(7'%',*/(6%'0 +8',

K1/p_{" G',' K /0 (+% 61:'5,6/.6112 .1+0'7 /( L) 6(7 /(7''7 /$ K /0 61:'5,6/.6112}

.1+0'7 /( L %&' ,/:&% /('EF61/%2 /0 0%,/.%"

H" ?'% iexp(K) = n) 062 a1, . . . , an/0 6 p4560' +$ K" G',' /0 6( ';6*-1' +$ 6 @(/%'12

:'(',6%'7 ';%'(0/+( L|K +$ %,6(0.'(7'(.' 7':,'' 1 =&',' K /0 61:'5,6/.6112 .1+0'7 /( L 6(7 iexp(L|K) = n &+170 ?'% L0= K(x)5' %&' ,6%/+(61 $F(.%/+( @'17 +8',

K" IF% L = L0(y1, . . . , yn−1)=/%& %&' 7'@(/(: 'EF6%/+(0

y₁p= a1xp+ a2 , yp2= a2xp+ a3 , . . . , yn−1p = an−1xp+ an .

A&'( trdeg(L|K) = 1) K /0 61:'5,6/.6112 .1+0'7 /( L 6(7 x, y1, . . . , yn−1 /0 6

p4560' +$ L|K" !

! _{!""# $% &'( )*%+,*+( "# -.-/0 1( '*2( LK}1/p_{= K}1/p_{(x). 3'( 45*6( Ω}

L"# 7$8(!(%&$*)4 "#

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xpda1+ da2= 0, xpda2+ da3= 0, . . . , xpdan−1+ dan= 0 .

1( 4((0 3'( 6*%"%$6*) <*5 ΩK =PνK daν → ΩL $4 $%=(6&$2( >9,& L ⊗KΩK → ΩL '*4 *

?(!%() "# 6"7$<(%4$"% -@A 4" K $4 *)+(9!*$6*)): 6)"4(7 $% L. B,!&'(!<"!( dx, dy1, . . . , dyn−1 $4

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δ(a + b) = δ(a) + δ(b) , δ(a_{· b) = a · δ(b) + b · δ(a)} (a, b_{∈ K).}

89%3 :% 9";% &9% <=-&/%3& >=)% δa

b 

= b−2_{· (b · δ(a) − a · δ(b))} (a, b_{∈ K, b 6= 0)}

"3* &9% 2-:%> >=)% δ(an_{) = na}n−1_{δ(a).-> n ∈ N >%02+ n ∈ Z /. a 6= 0+ ?- .-> "}

2-)63-1/") f(x) =Pνaνxν :/&9 aν, x∈ K :% 7%& δ(f (x)) = δ n X ν=0 aνxν  = n X ν=1 ν aνxν−1δ(x) + n X ν=0 δ(aν)xν =: f′(x)δ(x) + fδ(x) .

, *%>/;"&/-3 -. K ;"3/09%0 -3 &9% 2>/1% (%)* K◦ -. K+ ?- &9% *%>/;"&/-30 -.

K /3&- V .->1 " K◦40='02"5% Der(K, V ) -. HomK◦(K, V )+ @. char K = p > 0

" *%>/;"&/-3 ;"3/09%0 -3 &9% 0='(%)* Kp_{A %02%5/"))6 " 2%>.%5& (%)* -. 2>/1%}

59">"5&%>/0&/5 9"0 3- *%>/;"&/-3 6= 0+

'# -.)$+,&*+ */ 0$%&'()&*+, &+ ,&123$ $.)$+,&*+,4 *+ L|K !, - ,".-(-/#" #0"/(!1 2"#$ "34"5,!'5 4&"5 5) $"(!64!'5 δ '+ K !54' 5 L7,.1" V &, -85!98" .('#'50-4!'5 4' L: *+ a ∈ L ;!4& f = MinPol(a|K)< 4&"5

f (a) = 0 =_{⇒ f}′(a) δ(a) + fδ(a) = 0 =_{⇒ δ(a) = −}f

δ_(a)

f′_(a) .

*+ L = K(α) !, -5 !5,".-(-/#" "34"5,!'5 '+ $"0("" p ;!4& αp _{= a}

∈ K 4&"5 -$"(!6-4!'5 δ '+ K !54' -5 L7,.-1" V 1-5 /" "34"5$"$ 4' L != δ(a) = 0> *5 4&!, 1-," 4&" 1&'!1" '+ δ(α) !, +("" -5$ $"4"(?!5", 4&" "34"5,!'5> *+ L = K(x) !, .8("#) 4(-5,1"5$"54-# 4&"5 -5) $"(!6-4!'5 '+ K !54' -5 L7,.-1" 1-5 /" "34"5$"$ 4' L< 4&" "34"5,!'5 !, $"4"(?!5"$ /) 4&" !?-0" δ(x) '+ x ;&!1& !, +("" 4' 1&'',"> 5# #$/&+&)&*+4 $%& L|K '% " (%)* %B&%30/-3+ , !"#$%&#'( ') L|K C-> " K* !"#$%&#'( ') L# /3&- "3 L402"5% V /0 " *%>/;"&/-3 -. L /3&- V :9/59 ;"3/09%0 -3 KD %<=/;")%3&)6E /0 " K4)/3%"> 1"2 δ : L → V 0=59 &9"&

δ(a_{· b) = a · δ(b) + b · δ(a)} (a, b_{∈ L).}

!"#$%&'(') *'+')

!"#$%& !"# L = K(x1, . . . , xn) $" % &%#'()%* +,)-#'() ."*/ () n ')/"#"&0'1

)%#"2 (3"& K4 56") #6" 7%&#'%* /"&'3%#'3"2 ∂

∂x1

, . . . , ∂ ∂xn

%&" L1*')"%&*8 ')/"7")/")# /"&'3%#'()2 (+ L|K9 ') +%-# % $%2'2 (+ #6" L127%-" DerK(L, L)4 !" !"#$%& #$% !%&'()(*'+ (' ," ,'! -" -,../ *0%. )* -*112),)(0% .('3+ K ,'! %4)%'+(*'+ L|K *5 -*112),)(0% .('3+6 75 R (+ ,' (')%3.,8 !*1,(' 9()$ :2*)(%') &%8! K ,'! V , K;+<,-% )$%' ,'/ !%.(0,)(*' 5.*1 R )* V $,+ , 2'(:2% %4)%'+(*' )* K 0(, )$% :2*)(%') .28% +2-$ )$,) 9% -,' (!%')(5/ Der(R, V ) = Der(K, V ) . =2) )$%.% 1,/ >% *)$%. !%.(0,)(*'+? 5*. R = Z[i] %636 )$% !%.(0,)(*'

δ : R_{→ F}2= R/(1 + i) , δ(a + bi) = b mod 2 .

%" '$()(*+,+(-& ,'( L|K -' . /'"0 '1('23&425

(6 6# char K = 0 .20 &# (xi)i∈I &3 . ().237'20'27' -.3' 4# L|K8 (9'2 ':')+

K$0')&:.(&42 δ : L → V &3 0'(');&2'0 -+ (9' &;.<'3 δ(xi) #4) i ∈ I .20

(9' :."!'3 δ(xi)7.2 -' 7943'2 #)''"+ &2 V 5

((6 6# char K = p > 0 .20 &# (xi)i∈I &3 . p$-.3' 4# L|K8 (9'2 ':')+ K$0')&:.(&42

δ : L_{→ V &3 0'(');&2'0 -+ (9' &;.<'3 δ(x}i)#4) i ∈ I .20 (9' :."!'3 δ(xi)

"# K $% &'( )*$+( ,(-. /# L 0( 0*$&( ΩL $1%&(2. /# ΩL|K 21. 32-- $& &'( %)23(

/# !"#$#%&!'() *+ L4 5'(1 Der(L, V ) = HomL(ΩL, V )4

67 !"#$%&'$(!" ) 89:'-(*7; </* (23' a ∈ L #/*+ 21 (-(+(1& da= 6>$-. &'( L?%)23( 0$&' 62%$% (da)a∈L 21. .$@$.( />& &'( 1(3(%%2*A *(-2&$/1% &/ +2B( &'(

+2) a 7→ da $1&/ 2 K?.(*$@2&$/1 /# L;

Ω_L|K = M

a∈L

Lda.d(a+b)_{−da−db, d(a·b)−a·db−b·da, dc ; a, b ∈ L, c ∈ K}

37 !"#$%&'$(!" * 8C*/&'(1.$(3B7; D(& µ : L ⊗KL→ L 6( &'( +>-&$)-$32&$/1

a⊗ b 7→ ab #/* a, b ∈ L4 5'(1 I = Kern µ $% 2 +2E$+2- $.(2- $1 L ⊗KL0$&'

L_⊗KL/I≃ L4 F( .(,1( &'( L?%)23( /# .$G(*(1&$2-% /# L|K 2%

Ω_L|K = I/I2 0$&' &'( >1$@(*%2- .(*$@2&$/1 a7→ da := a ⊗ 1 − 1 ⊗ a

.7 +,-.%/0 H(,1$&$/1 21. 3/1%&*>3&$/1% I 21. 32**A $++(.$2&(-A /@(* &/ 21A (E&(1%$/1 L|K /# 3/++>&2&$@( *$1J% 0$&' ΩL = ΩL|Z·14 "# R $% 21

$1&(J*2-./+2$1 0$&' K>/&$(1& ,(-. K 0( J(&

ΩK= K⊗RΩR ,

6>& &'( (E2+)-( $1 I4IL4. %'/0% &'2& ΩR6= 0 21. ΩK = 0$% )/%%$6-(4

(7 1%!2!#($(!"0 *"+ L = K(a1, . . . , an)," - ./!+"#) 0"/"(-+"$ "1+"/2!'/ 3!+&

$"./!/0 "45-+!'/2

fµ(a1, . . . , an) = 0 6'( 1 ≤ µ ≤ m (fµ∈ K[X1, . . . , Xn]) ,

!7"7 L = Quot(R) 3!+& R = K[X1, . . . , Xn]/(f1, . . . , fm)7 %&"/

Ω_R|K = n M ν=1 Rdaν .D_Xn ν=1 ∂fµ ∂Xν (a1, . . . , an)· daν; 1≤ µ ≤ m E -/$ 2!8!#-(#) 6'( ΩL|K = L⊗RΩR|K7 #7 !"#$%&'

!" #$ char K = 0 %&' (xi)i∈I() % *+%&),-&'-&,- .%)- /$ L|K *0-& ΩL|K=Li∈ILdxi1

2/+- 3-&-+%445 6- 0%7- $/+ xi∈ L

x1, . . . , xn %43-.+%(,%445 (&'-8-&'-&* /7-+ K

! !"#$%&'(') *'+')

! "# char K = p > 0 $%& (xi)i∈I '( $ p)*$(+ ,# L|K -.+% ΩL|K =Li∈ILdxi/ 0,1+

2+%+1$334 5+ .$6+ #,1 xi∈ L7 x1, . . . , xn p)'%&+8+%&+%- ,6+1 K ⇐⇒ dx1, . . . , dxn L)3'%+$134 '%&+8+%&+%- '% ΩL|K 9(8+:'$334 5+ .$6+ dx = 0 '% ΩL|K ⇐⇒ x ∈ LpK . ;! "# L|K '( (+8$1$*3+ $32+*1$': -.+% ΩL|K = 0! "# !""#$ ,'( L|K -' . /0&('"+ 1'0').('2 '3('04&50 5# /'"246 $% L|K &4 4'7.).-"' &8 dimLΩL|K = trdeg(L|K)6

% 9:' ;&0&;." 0!;-') 5# 1'0').(5)4 ν(L|K) 5# L|K &4 1&<'0 -+

ν(L|K) =     

dimLΩL|K &# L|K &4 7!)'"+ ().04='02'0(."

5) &04'7.).-"'

1 + dimLΩL|K '"4'.

&# %&'(')*+*',$ ,'( L|K -' . /'"2 '3('04&50 .02 char K = p > 0 '()* char K = 0+*)+,*-.,/ 0.#1 0..#1 0.2# &)34 53657/86 9:' #5""5>&01 7)57')(&'4 .)' '?!&<."'0(@

0.# L|K &4 4'7.).-"'6

0..# A<')+ 2')&<.(&50 δ : K → V #)5; K &0(5 .0 L$47.=' V =.0 -' '3('02'2 (5 . 2')&<.(&50 5# L &0(5 V 6

0...# p$&02'7'02'0( '"';'0(4 &0 K )';.&0 p$&02'7'02'0( &0 L6 0.2# 9:' =.050&=." ;.7 L ⊗KΩK → ΩL &4 &0B'=(&<'6

-!"#&.$ 9:' &0B'=(&<&(+ 5# ΩK → ΩL &;7"&'4 (:.( 05 '"';'0( &0

LrK&4 7!)'"+ &04'7.).-"' 5<') K6

.# /0#1+ )!23!,1!)$

$% ,'( L|K -' . /'"2 '3('04&506 9:'0 >' :.<' .0 '3.=( 4'?!'0='

L⊗KΩK −→ ΩL −→ ΩL|K −→ 0

>:')' (:' /)4( ;.7 &4 &0B'=(&<' &8 L|K 4'7.).-"'6

% ,'( (R, m) -' . "5=." 25;.&0 =50(.&0&01 . /'"2 K >&(: )'4&2!' /'"2 L =

R/m6 9:'0 >' :.<' .0 '3.=( 4'?!'0='

m/m2 d

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&6 5/+%+-'*'+,4 8"0 x1, . . . , xr/" !1 F 9!0& #!1".(#) !1$"-"1$"10 $!:"("10!.#, dx1, . . . , dxr !1 ΩF |K7 %&"1 x1, . . . , xr.(" .#;"/(.!5.##) !1$"-"1$"10 '6"( K .1$ F |K(x1, . . . , xr)!, ,"-.(./#"7 !""#$ %&' (!)* +,-./ #",,"0) #!"/ 12132+242 5.6+' dx1, . . . , dxr!' 7!* "# -8-)' #"! Ω_{F |K} *&' )'+"69 +,-./ #",,"0) #!"/ 121:2&2 %6 6+/+&&)/74 8"0 d ∈ N0 .1$ E = K(x1, . . . , xn)/" . ,</2"#$ '+ F 9!0& trdeg(E_{|K) = dimF} n X ν=1 F dxν  ≥ d (_∗)

9&"(" 0&" $!:"("10!.#, dxν .(" 0.="1 !1 ΩF |K7 %&"1 F |E !, ,"-.(./#"7

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! !"#$%&'(') *'+')

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Ec Ec(y) EcEc′ E′

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K K(y + c′_{x) E} c′

:); (#/ E′ _{'# /7# "#$%&%'(# %(0#'&% 1 1()"5&# )- K(x, y) ! F 3 </ 1)!/% !"}

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/7# ( !#%& , "8) !/!#"" )- /7# Ec(y)-)(();"

c6= c′, Ec(y) = Ec′(y) =⇒ E_c(y) = K(x, y) .

<- N " /7# !5?'#& )- ? ! ?%( $&)$#& "5'.#(," )- E′_{|K(x, y) /7#! /7#&# %&#}

%/ ?)"/ N #(#?#!/" c ∈ K× _{; /7 E}

c(y) 6= K(x, y)3 6)& %(( )/7#&" ;# 7%*#

Ec(y) = K(x, y)+ ") Ec= K(y + cx)+ ") F |K(y + cx) &#05(%&3

"# !"!##$"%& *"+ x, y ∈ F ," -#.",(-!/-##) !0$"1"0$"0+ '2"( K -0$ dx 6= 0 '( dy 6= 0 !0 ΩL|K3 *"+ u ," -0 !0$"+"(4!0-+" '2"( F -0$ K1 = K(u)5

F1 = F (u)3 %&"0 F1 !6 (".7#-( '2"( K1(y + ux)3

!""#$ %&'() K1(y +ux) = K1(x+u−1y)*) +,- ,../+) dx 6= 00 1)23,(&'4 K

5- eK *) (,' ,../+) 67,6 K &. &'8'&6)0 F1|K1&. !)4/3,!0 9- 67) 2!"2".&6&"'

&' ,: 67)!) ,!) "'3- 8'&6)3- +,'- );()26&"',3 c ∈ K1./(7 67,6 F1|K1(y + cx)

&. '"6 !)4/3,!0 <# u *)!) );()26&"',3= ,33 u+a *&67 a ∈ K *"/3> 5) );()26&"',3 5- ,223-&'4 67) ,/6"+"!27&.+ σa: u7→ u + a "# F1|F *7&(7 +,2. 67) 8)3>.

K1= K1(x, y),'> E1′ = E′(u)"# 67) 2!""# "# 2!"2".&6&"' ,: "'6" &6.)3#0 ?7&.

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*"+ U = (uiν)i=1...d, ν=1...n ," !0$"+"(4!0-+"6 '2"( F 5 #"+ K1 = K(U )5 F1 =

F (U )-0$ yi= ui1x1+· · · + uinxn 9'( 1 ≤ i ≤ d3 %&"0 F1|K1(y1, . . . , yd)

!6 (".7#-(3

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