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 ten questions have about the same number of marks.

1.

Find integers m and n such that 21m+50n = 1.

2.

Let F be a field. Prove that there is an infinite number of irreducible polynomials in F[x].

3.

Let G be the dihedral group of order 10 consisting of all symmetries of a regular pentagon (with 5 sides). Find the conjugacy classes in G.

4.

Show that there is an infinite number of integers a such that x⁵+6x+a is irreducible over the rational numbers. Show that there is an infinite number of integers a such that this polynomial is reducible over the rational numbers.

5.

If K is a field containing a field F such that the degree [K:F] is odd, show that K has no subfield of degree 2 over F. Show that Ö2 is not contained in any extension of the rational numbers of odd degree.

6.

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

(a)

x³-x-1 over the field F₃ with 3 elements.

(b)

x⁴+6x-4 over the field Q of rational numbers.

(c)

x⁴+8x²+6 over the field Q of rational numbers.

7.

Find a solution in the field F₅ with 5 elements of the equations

8.

Let F be a field of characteristic p ¹ 0 and let n be a positive integer. Show that (a+b)^p = a^p+b^p for all a,b Î F. Show that (a+b)^pn = a^pn+b^pn for all a,b Î F. Show that the set of all roots of x^pn-x = 0 in F is a finite subfield of F.

9.

Show that each of the following complex numbers are algebraic numbers, and determine their degrees over the rational numbers Q.

(a)

Ö3+Ö7

(b)

cos(2p/23)+isin(2p/23)

(c)

[((1+i))/(Ö2)]

10.

Prove that the number of elements of any finite field is a power of a prime number.