Tutorial on Differential Galois Theory I

pages

English

Documents

Écrit par
T. Dyckerhoff

Publié par
Vofeg

Lire

Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres

pages

English

Documents

Lire

Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres

Publié par

Vofeg

Langue

English

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 ...

Voir

Publié par

Vofeg

Langue

English

Tutorial on Differential Galois Theory

T. Dyckerhoff

Department of Mathematics University of Pennsylvania

02/12/08 / Oberﬂockenbach

−x

x)2dZR2=(Z−xRed12x22xd12x−e2x−0Z∞0=Z2πbinibyFuocralopθdrdr2r−e

Motivation

πs=tenadior

RZ=x−12x−e2Z00∞Z=π2drdr−e2r1dx222dxbinibyFu

(

e−x2

πs=

x2 −

Motivation

dx)2

idroetanlopθocra

Motivation

(Zx2dx)2= − e R

Zbae−x2dx

ZR2e−x12−x22dx1dx

by Fubini

==Z0π2πZ0∞e−r2rdrdθpolarcoordinates

Motivation

Zbe−x2 dx a

= ?

(Ze−x2dx)2=ZR2e−x12−x22dx1dx2 R =Z2πZ0∞e−r2rd rdθ 0

by Fubini

polar coordinates

=π

Motivation

Zabe−x2dx

= ?

(ZRe−x2dx)2=ZR2e−x12−x22dx1dx2 2π∞ =Z0Z0rdθ e−r2rd =π

by Fubini

polar coordinates

Motivation

−x

elementar

function?

oHwysmmtercisimy00+2xy0=0?

Motivation

How

− ex2dx

an elementary function?

symmetric is

00+2xy0=0

Motivation

2 − exdx

an elementary function?

How symmetric is

y00+2xy0=0

ebtsehysmmteirse(i,4√2)/Q)descri

Evariste Galois asked:

Galois theory of polynomials

the

polynomial

−

symmetric

How

teirsera2)2=ySmmac)2=(bdcd)2=−2(cb(=(=2)(=2)2)dansioab:(2R4√atel,2i−−,√44i2√√4,2ots:ofroSetethegroupG=Aut(Qletaoisnhtrefero)/√2ncQeesodlralttilﬁgniQdle4,i(braialgey:spcallllerceatno.salitswotrohesprechhiatumrepetfosnoit

saietrmeymSd)Q/rcse,i(Q)2√4meymietresibesths

Evariste Galois asked:

Galois theory of polynomials

Set of roots:√42,i4√2,−√2,−i4√2 4 Relations: (ab)2= (ad)2= (bc)2= (cd)2 (ac)2= (bd)2=2

How symmetric is the polynomialX4−2?

=−2

pereutrmioatofnsrehtstoocihwserhpectallrelationsa.glbearcilayls:ngtiitpli,Q(ldﬁeneQ/)2√4llasedoctionrelarefosthegeorerhtuA(tpu=G

Voir