Last Name^{Name}  ^{Student No}^{Department}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Instructor}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{Basic Algebraic Structures}^{Final}^{Math 116}^{2005-2006}^{Spring}^{AB, SF, TÖ}^{22.05.2006}^{13.30}^{120 minutes}  ^{5 Questions on 5 Pages}^{Total 80 Points}^{1}  ^{2}  ^{3}  ^{4}  ^{5}

READ THE PROBLEMS CAREFULLY AND GIVE DETAILED WORK 1. (6+6+4 pts.) Let G be cyclic group of order 15 with a generator a a) Find other generators of G.

b) Find all subgroups of G.

c) What is the order of a^{3} ? a^{7} ?

2. (4+4+8 pts.) Consider element a = (123)(45) in S_{5}.

a) What is the order of the cyclic subgroup ⟨a⟩ ⊂ S_{5} generated by a ?

b) What is the index of the subgroup ⟨a⟩ in S_{5} ?

c) Let H = ⟨a⟩ and b = (1, 2) ∈ S_{5}. Give a list of the elements of b H (the left coset of b for subgroup H ⊂ S_{5}).

3. (5+7+6 pts.) Let R = {[0], [2], [4], [6], [8]} ⊂ ℤ_{10}.

a) Show that R is a subring of ℤ_{10}

b) Is R an integral domain ? Prove your answer. (Hint: check the multiplication table of R.)

c) Is R a field ? Prove your answer.

4. (5+5+5+3 pts.) Either prove that polynomial f(x) = x^{3} + 3x + 2 is irreducible as an element of F [x], or (if it is reducible) determine the degrees of its prime factors, for the following fields F.

a) F = ℤ_{2}.

b) F = ℤ_{5}.

c) F = ℚ.

d) F = ℂ.

5. (6+6 pts.)

a) Find a monic polynomial f(x) ∈ ℤ_{5}[x] such that f(a) = 0 for all a ∈ ℤ_{5}.

b) Prove that if field F is infinite then F [x] contains no monic polynomials f(x) such that f(a) = 0 for all a ∈ F.