課程名稱︰密碼學
課程性質︰數學系選修
課程教師︰陳君明
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/03/15
考試時限（分鐘）：30
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Name:_______ Student ID number:_________
s=__=2+"the last digit of your ID", 2≦s≦11
1.True/False problems: (Do not explain why. Write down T or F only.)
a) (3Z,+) is a subgroup of (Z,+)
b) 3Z is an ideal of Z
c) C[x] is a subring of C[x,y]
d) C[x] is an ideal of C[x,y]
2.Condiser the group G=(Z13*, ×mod 13)
a) s^-1 =____ (the multiplicative inverse of s)
b) o(s) =____ (the order of s)
c) [G:<s>] =____ (the index)
d) Explain why G is a cyclic group
3.Consider the group homomorphism f:(Z12, + mod 12) → (Z13*, ×mod 13)
defined by f(1) = s
a) f(2)=____
b) f(-1)=____
c) Is f an isomorphism? Why?
4. a) Is the ring Zs (={0,1,2,...,s-1}) an integral domain? Why?
b) Is the quotient ring Z[x]/<x^2-1> an integral domain? Why?
5.Consider the symmetric group S5:
┌1 2 3 4 5┐ ┌1 2 3 4 5┐
│ │。│ │ =______
└2 4 5 1 3┘ └2 4 5 1 3┘
┌1 2 3 4 5┐-1
│ │ = ______
└2 4 5 1 3┘
6.|GL2(Z11)| = _____ |SL2(Z11)| = ____
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