课程名称︰代数导论二
课程性质︰数学系大二必修
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）︰2016.04.28
考试时限（分钟）：110 mins
试题 :
ALGEBRA EXAM II
4/28 2016
Write your answer on the answer sheet. You should include in your answer every
piece of reasionings so that corresponding partial credit could be gained.
Denote ζn = exp(2πi/n) in Ｃ
PART I. True or false. Either prove the assertion or disprove it by a counter-
example (6 point each):
(1) Let V theabelian group generated by x, y, z, with the relation :
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 group.
(3) The polynomial x^5 + 5x + 5 is irreducible in Ｑ(2^(1/3)).
(4) The number 3^(1/3) in Ｒ is contained in Ｑ(2^(1/3)).
(5) Let R be a 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.
PATT II. For each of the following problems, give according a short proof or an
example (8 point each):
(1) Find a polynomial f(x) in Ｑ[x] such that f(√2+√5) = 0.
(2) Prove that ζ5 not in Ｑ(ζ7).
┌ 3 1 2 ┐
(3) Identify the abelian group present by the matrix │ 1 1 1 │.
└ 2 3 6 ┘
(4) Two abelian group 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 point 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
if and only if g(x) is irreducible over K.
(3) There are 6 isomorphic classes of abelian groups of order 72.