1. Z[i] a + b = p Z Z[i]/(a + bi) ∼= Z/pZ. (2005 2
2
113.)
Z[i]/(a + bi) 0, 1, 2, · · · , p − 1 a + b = p Z , (a, b) = 1. u, v ∈ Z av + bu = 1. (a + bi)(u + vi) = m + i, m ∈ Z.
c + di ∈ Z[i], c + di = d(u + vi)(a + bi) + (c − dm).
c + di = c − dm ∈ Z[i]/(a + bi). Z[i]/(a + bi) =
QED
{0, 1, 2, · · · , p − 1} ∼
= Z/pZ ()
2. R n V = R (n ) V [−, −] : [P, Q] = tr(P Q), ||P || = [P, P ] ( tr(C) DZ C Æ)
(1) V [−, −] DZ
(2) P, Q, P + Q = R. ||P Q|| ≤
2
2
2
n×n
T
2
√1 ||R||2 .
2
(2005
7.)
P = E = −Q, ||P Q||
(1) Remark.
2
2
= n,
||R|| = 0.
(i) [P, Q] = tr(P T Q) = tr(QT P ) = [Q, P ];
(ii) [kP + lQ, X] = k[P, X] + l[Q, X]
[P, P ] = 0 P = 0 (2) U U P U DZÆ U RU = U P U+U QU, ||(U P U)(U QU)||
= ||U (P Q)U|| = ||P Q||, ||R|| = ||U RU|| , P Æ. (iii) [P, P ] ≥ 0,
T
T
−1
T
2
T
T
T
2
P = diag λ1 , λ2 , · · · , λn , Q = bij n×n , λi ≥ 0, bij = bji .
1
T

λ1 b11 · · ·
λ1 b21 · · ·

(P Q)T = QT P =  ..
 .


λ1 b11 · · ·
λn b1n
 λ2 b21 · · ·
λn b2n 


..  , P Q =  ..
 .
. 
λ1 bn1 · · · λn bnn

λ1 b1n
λ2 b2n 

..  .
. 
λn bn1 · · · λn bnn
||P Q||2 = tr((P Q)T (P Q))
= λ21 b211 + λ22 b221 + · · · + λ2n b2n1
+λ21 b212 + λ22 b222 + · · · + λ2n b2n2
+···
+λ21 b21n + λ22 b22n + · · · + λ2n b2nn .


λ1 + b11
b12
···
b1n
 b21
l2 + b22 · · ·
b2n 


R=
,
..
.
.


.
.
bn1
bn2
· · · ln + b1n
||R||2 = tr(RT R)
= (λ21 + 2λ1 b11 + b211 ) + b212 + · · · + b21n
+b221 + (λ22 + 2λ2 b22 + b222 ) + · · · + b22n
+···
+b2n1 + b2n2 + · · · + (λ2n + 2λn bnn + b2nn ).
P Q λ
2λ b ≥ 0. (*) i ii
||R||4 ≥ (λ21 + b211 + b212 + · · · + b21n )2
+(λ22 + b221 + b222 + · · · + b22n )2
+···
+(λ2n + b2n1 + b2n2 + · · · + b2nn )2
2
(*)
i
≥ 0, bii ≥ 0,
≥ 2||P Q||2.
||P Q|| ≤
√1 ||R||2 .
2
#
3. F = Z/3Z. f (x) = x + 2x + 1 ∈ F [x] F
K, K[y] f (y) (2005 3
10.)
f (x) F [x] f (0) = 1, f (1) = 1, f (2) = 1, f (x) F [x]
f (x) DZ K = F [x]/ < f (x) >, < f (x) > f (x) f (y) = y 3 + 2y + 1 = [y − x][y − 2xy + (x2 − 1)]
= [y − x][y − (x − 1)][y − (x + 1)].
4. A = {a , a , · · · , a }, B = {b , b , · · · , β } (n ≥ m).
A B B A 1 B A DZ C · m!.
2 1. S = {(A , A , · · · , A ) | ∅ = A ⊆ A, A ∩ A = ∅, A = ∪ A }.
T A B 1
2
n
1
2
m
m
n
1
2
m
i
i
m
k=1
j
i
σ : T → S, f → (f −1 (b1 ), f −1(b2 ), · · · , f −1 (bm ))
DZ∀(A , A , · · · , A ) ∈ S, g(A ) = b ,
σ(g) = (A , A , · · · , A ), g ∈ T . DZ |T | = |S|.
|S| = m! · k, k = |S|, S = {(A , A , · · · , A ) | (A , A , · · · , A ) ∈ S.
1
1
2
2
m
i
m
1
2
m
3
1
2
m
i
k n m 1|A| = 5, |B| = 3 5 = 1 + 1 + 3, 5 = 1 + 2 + 2.
k = C C /2 + C C /2. |S| = 3! · 25 = 150.
2. I A B |I| = m . A
A B − {b } T = A ∩ A ∩ · · · ∩ A . |T | = m +(−1) C (m−1) +(−1) C (m−2) +· · ·+(−1) C 1 .
5. K V W V i = 1, 2, · · · , m. m = 1 V = ∪ W ,
V = ∪ W . α ∈ W − ∪ W β ∈ ∪ W − W . λα + β, λ ∈ K. λα + β ∈ W , W V β ∈ W , λ ∈ K, λα + β ∈ ∪ W .
|K| ≥ m, λ , λ λ α + β ∈ ∪ W , i = 1, 2,
α ∈ ∪ W , Remark. K, |K| = m. K V k , k ≤ m
K = Z , V = {(0, 0, ), (0, 1), (1, 0), (1, 1)} K 2 V 1
5
1
4
1
5
2
4
n
n
i
1
1
m
n
2
2
m
n
1
2
m−1
i
m−1
i=1
i
m−1
i=1
m
m−1
i=1
i
m
m−1
i=1
1
2
m−1
i=1
i
m
m−1 n
m
m
i=1
i
i
m
m
m
i
m−1
i=1
i
i
i
2
V = {(0, 0), (0, 1)} ∪ {(0, 0), (1, 0)} ∪ {(0, 0), (1, 1)}.
6. n n + 2 α (i = 1, 2, · · · , n+2),
[α , α ] < 0, ∀i = j. x α = 0. DZ
x , x , · · · , x , 1DZ x r x α + x α + · · · + x α = −[x α + x α + · · · + x α ].
i
1
n+1
i=1
j
2
i
i i
n+1
i
1
1
2
2
r
r
r+1
4
r+1
r+2 r+2
n+1 n+1
0 ≤ [x1 α1 + x2 α2 + · · ·+ xr αr , −xr+1 αr+1 −xr+2 αr+2 −· · ·−xn+1 αn+1 ] =
xi (−xj ) ri=1 n+1
j=r+1 [αi , αj ] < 0,
x DZ 0 = [α , 0] =
2DZ
,
xα]=
x [α , α ] < 0, [αn+2
n+1
i=1
i
i i
i
n+2
i
5
n+2