Exercices corrigés sur les structures algébriques - PCSI 2

E e ∗
e∗e=e(E, ∗)
E∗e x
e∗e=e e ∗x=x∗e=x x
x∗x=e(E, ∗)
y e ∗e=e e ∗x=x∗e=x
e∗y=y∗e∗y x ∗x=x x ∗x=y x ∗y y
x e x ∗x=y x ∗y=y∗x=e y ∗y=x
∗e x y z
e e x y z
x x y z e
y y z e x
z z e x y
∗e x y z
e e x y z
x x e z y
y y z e x
z z y x e
x y z
Unn
Uzn= 1 |z|n= 1 |z|= 1 R+
xn= 1 Un
1Unzn=z0n= 1 (zz0)n=znz0n= 1
zn= 1 1
zn= 1 U

∗0x∗
(y∗z) = (x∗y)∗z=x+y+z−xy −xz −yz +xyz
x∗y= 0 ⇔y=x
x−11R\{1}
∗(R,+) x7→ 1−x
1−(x∗y) = (1 −x)(1 −y)
x∗x∗ ··· ∗ x1−x∗x∗ ··· ∗ x= (1 −x)n
x∗x∗ ··· ∗ x= 1 −(1 −x)n
∗
x∗(y∗x) = (x∗y)∗z=x+y+z+xyz
xy +xz +yz 0x
−x
]−1; 1[ ∗x < 1y < 1
(x−1)(y−1) = xy −x−y−1<0−(1 + xy)< x +y1 + xy
−1<x+y
1 + xy −1< x −1< y x+y 1 + xy <1
th : R→]−1; 1[
th (x+y) = th (x)∗th (y)
◦f1f2f3f4f5f6
f1f1f2f3f4f5f6
f2f2f6f4f5f3f1
f3f3f5f1f6f2f4
f4f4f3f2f1f6f5
f5f5f4f6f2f1f3
f6f6f1f5f3f4f2
G
f1f1
f2◦f2◦f2=f6{f1;f2;f6}
G f4f5f6f1f2
{f1;f3}
G
G{f1;f4} {f1;f5}
G
∀x, y ∈R∗x
|x|×y |y|=xy |xy|
{−1; 1}R∗
+

θ7→ eiθ ei(θ+θ0)=eiθeiθ0U
2πZ={2kπ |k∈Z}
axa−1aya−1=a(xy)a−1axa−1=
e ax =a x =e
y
a−1ya
y7→ a−1ya
H1H2H1∩H2H1
H2∗x y H1H2x∗y
H1∩H2H1∩H2
G
CG={x∈G| ∀y∈G, x∗y=y∗x}=∩
y∈GCy
Cy={x∈G|xy =yx}CyCG
x y
E(xy)−1=xy (xy)−1=y−1x−1yx
LxLy)Lz=xL(yLz) = x+y+z−2
1x x −2
NR\{1}
Z[i√2] 0 1
C
1 (un) (−un)

m6un6M n −M6un6−m
1
+∞
∆∩
∆∅ ∩ E
AP(E) ∆ A∆A= (A\A)∪
(A\A) = ∅∩A E B ⊂E
A∩B=E A =B=E
∩A∩B=∅ ⇒ A=∅B=∅
AA
(P(F),∆,∩)
E
√2√3
√2×√2 = 2 3 √6
3−2 = 1
a+b√2 + c√3 + d√6 (a, b, c, d)∈R4
K K 0 1
√2×√6=2√3√3×√6=3√2
RK a +b√2 + c√3 + d√6
a0+b0√2 + c0√3 + d0√6aa0+ 2bb0+ 3cc0+ 4dd0= 1 ab0+ba0+ 3cd0+ 3dc0= 0
ac0+ca0+ 2bd0+ 2db0= 0 ad0+da0+cb0+bc0= 0
(a, b, c, d)
(x−3)(y−2) = 6
6 8 6 ×1 3 ×2
2×3 1 ×6−
{(9,3); (6,4); (5,5); (4,8); (2,−4); (1,−1); (0,0); (−3,1)}

x6
y6z x >41
x+1 y+1 z63 4
x= 3 1
x+1 y+1 z61x=y=z= 3
x= 2 y= 2 1
x+1 y+1 z>1y= 3
y= 4 (2,3,6) (2,4,4) y>5
x= 1
1−1
1
x= 1 y=−z y x =−1
y=z= 1
{(2; 3; 6); (2; 4; 4); (3; 3; 3); (1; y;−y)|y∈N∗}