Tutorial  on  Differential Galois Theory I
62 Pages
English
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Tutorial on Differential Galois Theory I

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Learn all about the services we offer
62 Pages
English

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TutorialonDifferential Galois Theory IT. DyckerhoffDepartment of MathematicsUniversity of Pennsylvania02/12/08 / OberflockenbachZ Z2 2 2−x 2 −x −x1 2( e dx) = e dx dx by Fubini1 22R RZ Z2π ∞2−r= e r dr dθ polar coordinates0 0= πMotivationZ b2−xe dx = ?aZ2 2−x −x1 2= e dx dx by Fubini1 22RZ Z2π ∞2−r= e r dr dθ polar coordinates0 0= πMotivationZ b2−xe dx = ?aZ2−x 2( e dx)RZ Z2π ∞2−r= e r dr dθ polar coordinates0 0= πMotivationZ b2−xe dx = ?aZ Z2 2 2−x 2 −x −x1 2( e dx) = e dx dx by Fubini1 22R R= πMotivationZ b2−xe dx = ?aZ Z2 2 2−x 2 −x −x1 2( e dx) = e dx dx by Fubini1 22R RZ Z2π ∞2−r= e r dr dθ polar coordinates0 0MotivationZ b2−xe dx = ?aZ Z2 2 2−x 2 −x −x1 2( e dx) = e dx dx by Fubini1 22R RZ Z2π ∞2−r= e r dr dθ polar coordinates0 0= πm00 0How symmetric is y +2xy = 0 ?MotivationZ2−xIs e dx an elementary function?mMotivationZ2−xIs e dx an elementary function?00 0How symmetric is y +2xy = 0 ?MotivationZ2−xIs e dx an elementary function?m00 0How symmetric is y +2xy = 0 ?√ √ √ √4 4 4 4Set of roots: 2, i 2,− 2,−i 2Relations:2 2 2 2(ab) = (ad) = (bc) = (cd) =−22 2(ac) = (bd) = 2Symmetries are permutations of the roots which respect allrelations.algebraically:√4splitting fieldQ(i, 2)/Q encodes all relations√4therefore the group G = Aut(Q(i, 2)/Q) describes thesymmetriesGalois theory of polynomialsEvariste Galois asked:4How symmetric is the polynomial X −2?Symmetries are ...

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Tutorial on Differential Galois Theory
T. Dyckerhoff
Department of Mathematics University of Pennsylvania
02/12/08 / Oberflockenbach
I
e
b
2
x
Z
a
x)2dZR2=(ZxRed12x22xd12xe2x0Z0=Z2πbinibyFuocralopθdrdr2re
=
dx
Motivation
?
πs=tenadior
RZ=x12xe2Z00Z=π2drdre2r1dx222dxbinibyFu
Z
Z
R
(
e
dx
b
ex2
πs=
x2
e
Motivation
dx)2
idroetanlopθocra
=
a
?
Motivation
(Zx2dx)2= e R
Zbaex2dx
=
ZR2ex12x22dx1dx
2
?
by Fubini
==Z0π2πZ0er2rdrdθpolarcoordinates
Motivation
Zbex2 dx a
= ?
(Zex2dx)2=ZR2ex12x22dx1dx2 R =Z2πZ0er2rd rdθ 0
by Fubini
polar coordinates
=π
Motivation
Zabex2dx
= ?
(ZRex2dx)2=ZR2ex12x22dx1dx2 2π=Z0Z0rdθ er2rd =π
by Fubini
polar coordinates
Motivation
Is
Z
e
x
2
dx
an
elementar
y
function?
oHwysmmtercisimy00+2xy0=0?
Motivation
Is
Z
How
ex2dx
an elementary function?
symmetric is
y
00+2xy0=0
?
m
Motivation
Is
Z
2 exdx
an elementary function?
How symmetric is
m
y00+2xy0=0
?
ebtsehysmmteirse(i,42)/Q)descri
Evariste Galois asked:
4
Galois theory of polynomials
the
is
X
polynomial
2?
symmetric
How
teirsera2)2=ySmmac)2=(bdcd)2=2(cb(=(=2)(=2)2)dansioab:(2R4atel,2i,44i24,2ots:ofroSetethegroupG=Aut(Qletaoisnhtrefero)/2ncQeesodlralttilgniQdle4,i(braialgey:spcallllerceatno.salitswotrohesprechhiatumrepetfosnoit
saietrmeymSd)Q/rcse,i(Q)24meymietresibesths
Evariste Galois asked:
Galois theory of polynomials
Set of roots:42,i42,2,i42 4 Relations: (ab)2= (ad)2= (bc)2= (cd)2 (ac)2= (bd)2=2
How symmetric is the polynomialX42?
=2
pereutrmioatofnsrehtstoocihwserhpectallrelationsa.glbearcilayls:ngtiitpli,Q(ldeneQ/)24llasedoctionrelarefosthegeorerhtuA(tpu=G