课程名称︰代数导论二
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）︰2016/6/30
考试时限（分钟）：120
试题 :
Part I (60 points) True or false. Either prove the assertion or disprove
it by a counter-examaple (10 point each):
(1) A group of order 63 is simple.
(2) Every group of order 121 is commutative.
(3) The Sylow 3-group of Z/6Z * Z/30Z is Z/3Z * Z/3Z.
(4) The Galois group of the splitting field of x^3+4x+1 over Q is
cyclic.
(5) Let V be the abelian group generated by x, y, z, with relations:
2x+3y+z=0, x+2y=0, 4x+y+3z=0. Then V is an infinite group.
(6) The polynomial x^5+5x+15 is irreducible in the field Q(√2).
Part II (75 points) For each of the following problems, give accordingly
a short proof or an example (15 points each):
(1) Find two Galois extensions of degree 4 with non-isomorphic
Galois groups.
(2) Show that the number of monic degree 5 irreducible polynomials
over F_3 is 28. 3 1 2
(3) Identify the abelian group presented by the matrix ( 1 1 1 ).
2 3 6
(4) Let R be an integral domain that contains a field F as subring
and that is finite dimensional when viewed as vector space over
F. Then R is also a field.
(5) Two abelian groups of order 2016 having the same amount of
elements of order 6 must be isomorphic.