Tutorial on Differential Galois Theory I

T. Dyckerhoff

Description

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 ...

Informations

• Publié par: Vofeg
• Nombre de lectures: 57
• Langue: English

Extrait

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) 2 d Z R 2 =( Z −x R ed 12 x 2 2 xd 1 2 x− e 2 x− 0 Z ∞ 0 = Z 2π binibyFu ocralopθdrdr 2 r− e

=

dx

Motivation

?

π s = tenadior

R Z = x− 1 2 x− e 2 Z 00 ∞ Z = π2 drdr − e 2 r 1 dx 2 2 2 dxbinibyFu

Z

Z

R

(

e

dx

b

e −x 2

π s =

x 2 −

e

Motivation

dx) 2

idroetanlopθocra

=

a

?

Motivation

( Z x 2 dx) 2 = − e R

Z b a e −x 2 dx

=

Z R 2 e −x 12 −x 22 dx 1 dx

2

?

by Fubini

= = Z 0 π 2π Z 0 ∞ e −r 2 rdrdθpolarcoordinates

Motivation

Z b e −x 2 dx a

= ?

( Z e −x 2 dx) 2 = Z R 2 e −x 12 −x 22 dx 1 dx 2 R = Z 2π Z 0 ∞ e −r 2 rd rdθ 0

by Fubini

polar coordinates

=π

Motivation

Z a b e −x 2 dx

= ?

( Z R e −x 2 dx) 2 = Z R 2 e −x 12 −x 22 dx 1 dx 2 2π∞ = Z 0 Z 0 rdθ e −r 2 rd =π

by Fubini

polar coordinates

Motivation

Is

Z

e

−x

2

dx

an

elementar

y

function?

oHwysmmtercisi m y 00 +2xy 0 =0?

Motivation

Is

Z

How

− e x 2 dx

an elementary function?

symmetric is

y

00 +2xy 0 =0

?

m

Motivation

Is

Z

2 − e x dx

an elementary function?

How symmetric is

m

y 00 +2xy 0 =0

?

ebtseh ysmmteirse (i, 4 √ 2)/Q)descri

Evariste Galois asked:

4

Galois theory of polynomials

the

is

X

polynomial

2?

−

symmetric

How

teirsera 2 )2= ySmm ac) 2 =(bd cd) 2 =−2 ( cb(=(= 2 )(= 2 ) 2 )da nsio ab : ( 2 R 4 √ atel ,2i−−, √ 44 i2 √√ 4 ,2ots:ofroSet ethegroupG=Aut(Q letaoisn htrefero )/ √ 2ncQeesodlralttilfigniQdle 4 ,i( braialgey: sp call ll er ceat no.salit swotrohesprechhiatumrepetfosnoit

saietrmeymS d)Q/rcse,i(Q)2 √ 4 meymietr esibe s th s

Evariste Galois asked:

Galois theory of polynomials

Set of roots: √ 4 2,i 4 √ 2,− √ 2,−i 4 √ 2 4 Relations: (ab) 2 = (ad) 2 = (bc) 2 = (cd) 2 (ac) 2 = (bd) 2 =2

How symmetric is the polynomialX 4 −2?

=−2

pereutrmioatofnsrehtstoocihwserhpectall relations a . glbearcilayl s : ngtiitpli,Q(ldfieneQ/)2 √ 4 llasedoctionrela refo s thegeorerhtuA(tpu=G