课程名称︰代数导论一
课程性质︰系必修
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）︰2015/11/26
考试时限（分钟）：130
试题 :
Write you answer on the answer sheet. You should include in your
answer every piece of reasonings so that corresponding partial credit
could be gained.
Part I. True or False. Either prove the assertion or disprove it by a
counter-example (6 point each):
(1) A ring of order 5 must be isomorphic to Z/5Z.
(2) √2 + √6 is an algebraic number.
(3) The (multiplicative) group of units of Z/12Z is a cyclic group.
(4) The polynomial x^2+11x+26 divides x^3+2x^2+x+1 in F5[x].
(5) Every nonzero ideal of the ring of Gauss integers contains a
positive integer.
Part II. For each of the following problems, give accordingly a short
proof or an example (8 points each):
(1) The kernel of the ring homomorphism C[x,y,z]→C[t] defined
by x |-> t, y|->t^3, z|->t^5 equals (x^3-y,x^5-z).
(2) Every ring of order 10 is isomorphic to Z/10Z.
(3) The ring Q[x]/(x^3+3) is actually a field.
(4) Find four rings, all of order 4, non-isomorphic to each other.
(5) A domain of finite order is a field.
Part III. Give a complete proof (10 points each):
(1) (a) Every finite group of isometry on R^2 has a fixed point.
(b) Let R be a ring, f(x)∈R[x] and denote R' = R[x]/(f(x)).
There is some α∈R' satisfying f(α) = 0.
(2) Show that Z[x]/(3,x+4) is isomorphic to F3.
(3) Show that Q[x]/(x^2-4) is isomorphic to Q x Q.