[PDF] Cours de M1 fondamental Algèbre et théorie de Galois

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A ???(A;)??? ?? ??????? ???????A=An f0g? ?? ??? ???A??? ?? ?????? u;v2Z???? ???ua+vb= 1??? ????? ??????r0:=jaj??r1:=jbj?? ??????r2?? ???a? ?? ???? ???????u;v???? ???up+va= 1? ????upb+vab=b? ?? ??? ?????? ???p??????b? p v11pv22pvrr=p01v01p02v02p0r0v0 r0?????p1??? ???? ? ?? ??? ?? ????? ???p0i??? ?????? ? v v ppp(n)? ??"(n)??????? ?? ????? ??n?? ?? ??????? ??? ?????? pgcd(n;m) =Y pp min(p(n);p(m))??ppcm(n;m) =Y pp max(p(n);p(m)): ?? ?? ?x0(x0+ 1) = 2y03? ??x0= 2x00??? ????? ?????x00??(x0+ 1)??????? ???? (1;0)??(3;2)? x

2+N=y3;??N2Z??? ???:

x

2+ 2 = (x+p2)(xp2)???? ????x2Z[p2]? ???? ??????? ??? ??? ?????? ???

?? ?????x=up11prr???? ???pi????P??u2A? x

22 = 4 = (1 +p3)(1p3):

u

Z[p3]?

??? ???? ??? ??????? ?? ??????? ???j:=1+p32 ???? ??????x+p3???? ?? ????? x+p3 =u(a+bp3)3=u((a39ab2) + (3a2b3b3)p3) ????u2 f1;1p32 a

39ab2+ 3a2b3b3= 2:

a

39ab2+ 3a2b3b3a3+ 3ab2+ 3a2b+b3(a+b)3(mod4):

x

23 = (1 +p5)(1p5):

?? ??? ???? ????Z[p3]? (a1;;an) = (a1) ++ (an) =Aa1++Aan S

1+S2=fx2A;9(s1;s2)2S1S2; x=s1+s2g:

I+J=fx2A;9(i;j)2IJ; x=i+jg:

IJ:=fx2A;9n2N;9(i1;;in;j1;;jn)2InJn; x=i1j1+injng: xy2I)x2I??y2I? (2) = (2;1 +p5)(2;1p5) = (2;1 +p5)2 (3) = (3;1 +p5)(3;1p5) (1 +p5) = (2;1 +p5)(3;1 +p5) (1p5) = (2;1p5)(3;1p5): p a ip ????? ??? ????Z[i]? ?p= 2??p1(mod4)? ? ????=a2+3b24 =p? ?? ?? ???? ???? X X V(f1;;fr) :=f(z1;;zn)2Cn; f1(z;;zn) ==fr(z1;;zn) = 0g: f C :O(V0)! O(V); f07!~'f:

O(Cn) =C[X1;;Xn]! O(V);f7!fjV

????f2 O(V)??k2N?fk= 0)f= 0? x

C[X1;;Xn]!A; Xi7!xi:

I a6= 0??ab=ac)b=c; X (a;a0) + (b;b0) := (a+b;a0+b0)??(a;a0)(b;b0) = (ab;a0b0):

Z=nmZ!Z=nZZ=mZ; a(modnm)7!(a(modn);a(modm))

e?

A!AeA(1e)?a7!(ae;a(1e))

Z=nZZ=mZ?

?8a;a02A; '(a+a0) ='(a) +'(a0)??'(aa0) ='(a)'(a0)? ?'(1) = 1?

A??? ???' = idA0?? '= idA?

( ; 0) :B!AA0; b2( (b); 0(b)) ' = 0? A

0??? ?? ????? ??A?

?? ?'(a0a) ='(a0)0 = 0? ????a0a2???(')??? '(I)???? ????? ??? ?? ?????? ??????(x+y)n=P k n pI:=x2A;9k2N;xk2I: ???x2A?? ?? ?????x=a1e1++arer??r2N? ???ei???? ????E? ?? ???ai???? ????A? ???????E=fx1;;xng? ?? ???? ????? ??? ????? (x1;;xn)?? ??????Ax1++Axn: ?????? ??A?? x2I? ???? ????? ??? ??????? ?? ?(x) +I=A? ???? ?? ???? ??????1 =ax+i????i2I ???(X;Y)????? ??????? ???? ????? ????? ??????? (0)(X1)(X1;X2) (X1;;Xn): ??IJ= (a1b1;;arb1;a1b2;;arb2;;arbs)

8a;b2A; ajb,(a)(b):

IJ=nmZ; I\J= ppcm(n;m)Z; I+J= pgcd(n;m)Z:

p= (2) + (1 +p5)??? ??????? ???????p2= (2)? x(modI)? ??????? ??x????A=I? ???? ??? ????? ???? ?????? ?? ???? ??????? ??? ????xx0(modI)??yy0(modI)? ?? ? ????i00= (iy+i0x+ii0)2I? xy1(modI)????yx=1??x??????? ???? ?? ??????? ????A=I? ?? ??????? ??? ????

I(p0) =

pI? A IA 0 A=I' ????i2I? ?? ? ????'(x0) ='(x) +'(i) ='(x)???????i2???(')? ????? ?????? ?? A=I? ??????? ???(') =1

I(???('))?????? ??? ???(') =1

B A=I ':AA=???(')!??('),!A0 J:A I!A=I

J!A=J:

(A=I)=(J=I)!A=J:

J????? ?????? ?? ??

??????? ???I(J)??? ???? ?? ??????? ?? ?1

I(???(

J)) =???(

JI) =???(J) =J?

J) =I(1

I(???(

I+JI????

(A=I)=((I+J)=I)!A=(I+J):

A=(I\J)(

I;

J)!A=IA=J:

I;

A=(I\J)!A=IA=J:

I;

J)?? ???? ??????? ??? ???? ????a;b2A? ?????

I; I;

Z=nmZ!Z=nZZ=mZ?

(a;m)7!am ???(a+a0)m=am+a0m??(aa0)m=a(am)? ????1m=m? ?? ??? ????? ???M??? ?? ??????? ???A?? (a)(m)? a':m7!a'(m);??'+'0:m7!'(m) +'0(m): B '!A!???Z(M) Hom

A(M;N)HomB(M;N):

???? ??????? ????? ????? ??A? M mm0,mm02N mm0)amam0??mm1;m0m01)(m+m0)(m1+m01)

MM=???( )!??( ),!M0:

M N!M=N

P!M=P:

(M=N)=(P=N)!M=N:

P) =N(???(P)) =N(P) =P=N?

N+P:=fm2M;9n2N;9p2P;m=n+pg

:N ,!N+P(N+P)=P

N=(N\P)!(N+P)=P:

N=(N\P)!(N+P)=P:

?(mi)i2I+ (m0i)i2I:= (mi+m0i)i2I ?a(mi)i2I:= (ami)i2I? Y i2IM Y i2IM i j!Mj: M i2IM i:=( (mi)i2I2Y i2IM i;mi= 0???? ??????? ????i) ??? ??I?? ?? ???? ??? ????? ?? ?????A??????? ??Q i2IMi=Q i2IMi? ???????I=f1;;rg?? ?? ???? ?????Lr j:Mj,!M i2IM i Hom A N;Y i2IM i! !Y i2IHom

A(N;Mi):

i2IMi!N??? ??? i= i Hom A M i2IM i;N! !Y i2IHom

A(Mi;N):

i2I i(mi)? ?? ??? ? ?? ???? ??????? i(Mi)?? ??????i? ?? ???? ??????? ??L

M(E) =fm=a1e1++arer; r2N;ei2E;ai2Ag:

X i2IM X i2IM i=fX i2Jm i; JI???;mi2Mig: i2IMi!M??????? ??? X i2IM i=??(): i2IMi!P i2IMi? m i2Mi? ?? ?P i2Imi= 0) 8i2I;mi= 0: ???? ?? ???? ??????i= 1;;r?? ?Mi\P j6=iMj=f0g? M i=A???? ????i2I? ?? ???? ????? A I:=Y i2IA??A(I):=M i2IA: A Hom

A(A(I);M)!MI:

m!M??? ??????1???m? ?? ??? ????? ???'m(a) :=am? (P i2Iaiei) =P iaimi= 0) 8i;ai= 0 ???? ?? ?????P i2Iaimi????ai2A???? ????i? i2Iaimi????ai2A???? ????i? ??? ??????? ???? ?? ????? ??? ??? ????? ???????0 = 2332? ? ??? ?????? ????? ??M????f1g??f1g? ? ??I??? ?? ????? ?????? ?? ??? ??? ??A? ????? ??A???????M:=A=I????? ??? ??????M? i=1aijei? ?? ??????? Hom

A(N;M)! Mn;m(A)

A=Z=M?? ???????f2;3g?

??????? ???? ????? ??????? ?????f2g?

M=AX+AY?

i;jaijei? ?? ???????P= (aij)i;j2 MnM(A)??? ?? ??????? ???? A m // m A n n (A=m)m //(A=m)n ???P??????? ?? ?????? ?? ??????rr??? ???? ??r= 0? ???? ???fi???? ???? ??F??? ???? ????k= 2;;r+ 1? ?????? ?????k?? ?????? ???((aij)16i6r;16j6r+1;j6=k? ????? ???? ????i= 1;;n? ?? ?????Pr+1 k=1(1)k+1aikk??? ?? ?????? ?? ??????r+ 1??P?? ?? k=1kfk= 0? ?? ?????16= 0? ?? ???? ??? ?? ???????F?? ????? A ?M?? ???? ??? ??NM)M=N?? ???? ???? ?N?? ???? ??? ??M=N?? ???? ???)M?? ???? ???? ???xn+1? ?? ???? ?????A??????? ??M??? ?? ???? ???? ???????M=S 1;;m r r+1;;p r+1;;p s????P? equotesdbs_dbs50.pdfusesText_50

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