Name: Student number: METU MATH 116, Makeup _{1 2 3 4 Σ}

Tuesday, June 9, 2014, at 13:40 (100 minutes), totally 60 points

Instructors: M.Bhupal, S.Finashin, F.Ozbudak, E.Solak Instructions: Please, be accurate and show clearly the logic of your solutions. Only the answers are not enough: indicate your calculations and arguments.

Problem 1. (15 pts) Solve a lnear system in ℤ_{29}

\begin{aligned}[7] & x + [2]y & = [1] \\ [12] & x − [1]y & = [2] \end{aligned}

Problem 2. (15 pts) Assume that G is a group and H ⊂ G a subgroup of index [G : H] = 2. Prove that H is neccessarily a normal subgroup of G.

Problem 3. (15 pts) Consider the ring of upper triangular matrices T_{2} = \{[\begin{matrix}a & b \\ 0 & c \end{matrix}] | a, b, c ∈ ℤ\}

and let I = \{[\begin{matrix}0 & x \\ 0 & y \end{matrix}] | x, y ∈ ℤ\}. Prove that I is a (2-sided) ideal in T_{2}.

Problem 4. (15 pts) Present polynomial f(x) = x^{4} + x^{3} + x^{2} + x as a product of irreducible polynomials over ℤ_{3}. (Explain, why your factors are irreducible).