课程名称︰代数导论一
课程性质︰必修
课程教师︰陈其诚
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/1/11
考试时限（分钟）：120
试题 :
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).