课程名称︰代数导论二
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）︰2016/4/28
考试时限（分钟）：120
试题 :
Write yor answer on the answer sheet. You should include in your
answer every piece of reasonings so that corresponding partial credit
could be gained. Denote ζ_n := e^(2πi/n) ∈ C.
Part I. True or false. Either prove the assertion or disprove it by a
counter-example (6 point each):
(1) Let V be the abelian group generated by x, y, z, with relations:
4x+2y+8z=0, 2x+16z=0. Then V is a finite group.
(2) The ideal (x,y) in R := Q[x,y] is a free R-module.
(3) The polynomial x^5+5x+5 is irreducible in Q( 2^(1/3) ).
(4) The number 3^(1/3) ∈ R is contained in Q( 2^(1/3) ).
(5) 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.
Part II. For each of the following problems, give accordingly a short
proof or an example (8 point each):
(1) Find a polynomial f(x) ∈ Q[x] such that f( √2 + √5 ) = 0.
(2) Prove that ζ_5 is not in Q(ζ_7). 3 1 2
(3) Identify the abelian group presented by the matrix ( 1 1 1 ).
2 3 6
(4) Two abelian groups of order 343 having the same amount of
elements of order 7 must be isomorphic.
(5) Find a basis for the Z-module of integer solutions of the system
of equations: x+2y+3z=0, x+4y+9z=0.
Part III. Give a complete proof (10 points each):
(1) Let α be an element of an extension K/F which is algebraic
over F, and let f be the irreducible(minimal) polynomial for
α over F. Show that the canonical map F[x]/(f) → F[α],
g(x) |-> g(α), is an isomorphism, and hence F(α) = F[α].
(2) Let α and β be complex roots of irreducible polynomials f(x)
and g(x) in Q[x]. Let K = Q(α) and L = Q(β). Prove that
f(x) is irreducible over L if and only if g(x) is irreducible over
K.
(3) There are 6 isomorphic classes of abelian groups of order 72.