課程名稱︰代數導論一
課程性質︰必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/1/11
考試時限（分鐘）：120
試題 :
Write your answer on the answer sheet. You should include in your
answer every piece of reasonings so that corresponding partial credit
could be gained.
For each of the following problems, give accordingly a short proof or
an example (10 points each):
(1) Find two groups A and B, both of order 4, but not isomorphic
to each other.
(2) Find four rings A, B, C and D, all of order 4, but not isomorphic
to each other.
(3) Let G be a group of order 35. If G contains normal subgroups
of order 5 and 7, respectively, then G is cyclic.
(4) Find a group G with two elements a and b, both of order 2,
such that ab is of infinite order.
(5) The ideal (x-1) is the kernel of the ring homomorphism
Q[x] -> Q that sends each f(x)∈Q[x] to f(1).
(6) The ring Q[x]/(x^2+1) is a field.
(7) The ring Q[x]/(x^2-1) is isomorphic to Q x Q.
(8) The ring Z[√-2] is a unique factorization domain.
(9) The ring Z[√-5] is not a unique factorization domain.
(10) The polynomial p(x) = x^2016 + 11x + 11 is irreducible in Q[x].
(11) In Z[√-1], 3+4√-1 and 5+12√-1 are relatively prime.
(12) Find all x∈Z such that x≡6 (mod 7), x≡3 (mod 9) and
x≡15 (mod 32).