Please make sure that your name is on everything you hand in.

This is a ``closed book'' exam. (Calculators are not allowed.)

Answer as many questions as you can.

All questions have about the same number of marks.

1.

Express the permutation (12345)(643125) as a product of disjoint cycles and find its order. Is it an even or odd permutation?

2.

Define N(z) for a complex number z by N(z) = z[`z]. Write N(z) explicitly in terms of the real and imaginary part of z, and prove that N(xy) = N(x)N(y). Write (a²+b²)(c²+d²) as a sum of two squares of polynomials in a,b,c,d. Express 1313 = 13×101 as a sum of two squares of integers.

3.

Construct a field with 169 = 13² elements.

4.

Which of the following polynomials are irreducible? Give reasons for your answers.

(a)

x³+x+1 over the field R of real numbers.

(b)

x³+x+1 over the field F₂ with 2 elements.

(c)

x⁴+x²+1 over the field F₂ with 2 elements.

5.

Find the greatest common divisor of the polynomials x²+1 and x⁵+1 over the field F₂ with 2 elements, and express it in the form