## Tutorial 6 : Cyclotomic Extensions

1. Determine

2. Determine a primitive element of a subfield of so that

3. Put Determine the degrees of and over

4. Put and Show that

5. Let be a primitive element modulo where is a prime. Thus Let Using the list show how to find the sum of powers of which determines a subfield of so that where

6. Suppose and for some integer Show that can be diagonalized. Prove that the matrix where and is a field of chracteristic satisfies and cannot be diagonalized if

7. Show that and deduce that

8. Show that .

9. Show that and deduce that the coefficients of satisfy for all

10. Let be a prime number and Show that
 if n is odd