# Tutorial on Differential Galois Theory II

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TutorialonDifferential Galois Theory IIT. DyckerhoffDepartment of MathematicsUniversity of Pennsylvania02/13/08 / OberﬂockenbachOutlineToday’s planPicard Vessiot ringsThe ∂ Galois group schemeThe Torsor theorem and applicationsDescent theory for Picard Vessiot extensionsn th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ... .. .    . . ..n−1 n−1∂ (y) −a −a −a ... −a ∂ (y)0 1 2 n−1⇒ We develop Picard Vessiot theory for general systems of1 st order equations:n×n∂(y) = Ay with A∈ Fwhich we denote by [A].Systems of ∂ equationsYesterday we considered:a ﬁeld F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 i⇒ We develop Picard Vessiot theory for general systems of1 st order equations:n×n∂(y) = Ay with A∈ Fwhich we denote by [A].Systems of ∂ equationsYesterday we considered:a ﬁeld F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 in th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ... . . .    . . ..n−1 n−1∂ (y) −a −a −a ... −a ∂ (y)0 1 2 n−1Systems of ∂ equationsYesterday we considered:a ﬁeld F with derivation ∂nan equation ∂ (y)+···+a ∂(y)+a y = 0 with a ∈ F1 0 in th order equation ⇒ a system of 1 st order equations:    y 0 1 0 ... 0 y    ∂(y) 0 0 1 ... 0 ∂(y)    ∂ =    . . ...

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##### Formal sciences

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Tutorial on Differential Galois Theory
T. Dyckerhoff
Department of Mathematics University of Pennsylvania
02/13/08 / Oberﬂockenbach
II
Outline
Today’s plan
Picard-Vessiot rings
The-Galois group scheme
The Torsor theorem and applications
Descent theory for Picard-Vessiot extensions
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an
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n
a ﬁeldFwith derivation
Systems ofoitauqe-sn Yesterday we considered:
.]A[yb
with
ai
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+
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+
(y
+
)
AhFwyti)yA=:s(notewedehichn×nw
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uationahordereqnt-:nsioatqureedrots-1fometsys10..0=(1)y.n)yy(
elopedevWtohtseisdrV-iPacsyalerenrgforyeoredrots-1fosmetsqeauitno:s()yA=ywithAFn×nwhichedewetonA[yb
n-th order equationa system of 1-st order equations: 0 0 (yy) 0 0 (yy) = n.1(y)a0an1 n.1(y )
.]
1 0 . a1
Systems ofnsuqe-oita Yesterday we considered: a ﬁeldFwith derivationan equationn(y) +∙ ∙ ∙+a1(y) +a0y=0 withaiF
. . . . . . . . . . . .
0 1 a2
Systems ofionsquat-e Yesterday we considered: a ﬁeldFwith derivationan equationn(y) +∙ ∙ ∙+a1(y) +a0y=0 withaiF n-th order equationa system of 1-st order equations: (yy) 100010......00 (yy)= n.1(y)a0a1.a2......an1.n1(y) We develop Picard-Vessiot theory for general systems of 1-st order equations: (y) =AywithAFn×n which we denote by[A].
)R(nY(::xiLGY[Y=F,dijAY)=dRanR2sinaniteY()]1main3R/Ftegraldo.e.i,cirtemoegsiwcnenoas)h(RotQu-isR4sinastnotson-t.non,i.empleDenitionAPcira-deVssoirtni]i[Aorgfngri-sa1htiwF/RnegsiF/RedbyeratdameafunosultnlaamrtitnodraciPfotoisseV-ffdoel=nsioctraV-seacdredlistoal-rivilsPiidea
a
system
of
-equations
ioit
Picard-Vessiot rings
Given
n
n×n
a
-ﬁeldFwith ﬁeld of constants
K
A
[A],
F
yhseetdryasednringcoincideswit
Deﬁnition APicard-Vessiot ringfor[A]is a-ringR/Fwith 1R/Fgenerated by a fundamental solution matrix:is YGLn(R) :(Y) =AYandR=F[Yij,det(Y)1] Ris an integral domain R/Fis geometric, i.e. Quot(R)has no new constants Ris-simple, i.e. no non-trivial-ideals
Given a-ﬁeldFwith ﬁeld of constantsK a system of-equations[A],AFn×n
2 3 4
no
Picard-Vessiot rings
initdsedaysterthyeeswidlforfcatle=ded-VessioPicariocgdicnoissnirtaricVed-ontifPso
Picard-Vessiot rings
Given a-ﬁeldFwith ﬁeld of constantsK a system of-equations[A],AFn×n
Deﬁnition APicard-Vessiot ringfor[A]is a-ringR/Fwith 1R/Fis generated by a fundamental solution matrix: YGLn(R) :(Y) =AYandR=F[Yij,det(Y)1] 2Ris an integral domain 3R/Fis geometric, i.e. Quot(R)has no new constants 4Ris-simple, i.e. no non-trivial-ideals
Picard-Vessiot ﬁeld= ﬁeld of fractions of Picard-Vessiot ring coincides with yesterday’s deﬁnition
y00
evogt(Rr
+y=0
Example overR(t)
)
2
order equation translates into the system yy21=0101 yy12
the
2-nd
s(cosit)nbve=yYc)t(t(so)t(nnismentalsoafundartxisiigulitnoamaPis=1)2(tin+s)2nirtoisseV-draci(t),[cosR(t))R=sot(tich)tw]is(n
=Rc[so(R)tsin((t),ithct)]ws+2)t(so1=2)t(niaricaPisiossVed-rtniogevRrt()
+y=0
Example overR(t)
the 2-nd order equation translates into the system yy12=1010 yy21
y00
a fundamental solution matrix is given by cossni((tt))csonis((tt))
Y=
Example overR(t)
y00
+y=0 the 2-nd order equation translates into the system yy2=0101 yy121
a fundamental solution matrix is given by Y=cos(t))csonsi((tt))sin(t
R=R(t)[cos(t),sin(t)]with
cos(t)2+sin(t)2=1
is a Picard-Vessiot ring overR(t)
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