課程名稱︰ 應用代數
課程性質︰ 選修
課程教師︰ 陳文進
開課學院： 電資院
開課系所︰ 資工系
考試日期（年月日）︰2017/05/09
考試時限（分鐘）： 150
試題 :
(用 Latex 格式)
1. Find all the positive integers m that make the following congruence
true simutaneously:
64 ≡ 13 mod m
2^8 ≡ 1 mod m
2. Find all integral solutions of the equation:
858x + 364y = 78
3. Find all integral solutions of the congruent equation:
56x ≡ 88 mod 96
4. Find the smallest positive integer that satisfies the following congru-
ences simutaneously:
x ≡ 5 mod 7
x ≡ 7 mod 11
x ≡ 3 mod 13
5. Let S = {a_1a_2a_3a_4 | a_i \in {0,1}, 1≦i≦4}be the set of binary code
words of length 4. We define the binary operator ⊕ on S as follows:
u = a_1a_2a_3a_4, v = b_1b_2b_3b_4, then u⊕v = c_1c_2c_3c_4 where
c_i = a_i + b_i mod 2 (1 ≦ i ≦ 4).
(a) Show that (S,⊕) is a group. Give the indetity and the inverr of
u = a_1a_2a_3a_4.
(b) Show that H = {0000,1010,0101,1111} is a subgroup of S.
(c) Find the coset of H in S.
6. G is a group. The center of G is Z = {u \in G |\forall x \in G, ax =xa}.
Prove that Z is an normal subgroup of G.
7. G is a group. If o(a) = 2 for all a \in G, a ≠ e, prove that G is a
communtative (Abelian) group.
8. G is a commutative group. a,b \in G, o(a) = m, o(b) = n. Find the order
of a^nb^m.
9. Let R[x] = {a_0 + a_1 x + ... + a_n x^n | a_i \in R. n ≧ 0} be the set
of polynomials with real coefficients, C = {a + bi| a,b \in R} be the
set of complex numbers.
(a) Show that the mapping φ: R[x] → C ,φ(p(x)) = p(i) is a sur-
jective homomorphism from additive group (R[x], +) to additive
group (C,+).
(b) What is Ker(φ) ?
(c) What is the quotient group R[x]/Ker(φ)?
(d) The Fundamental Theorem of Group Isomorphism tells us that
R[x]/Ker(φ) \simeq C. Given an isomorphism φ':
R[x]/ker(φ) → C.