Due Dec 17.

1. Two way choice: Exercise 8.9 or 8.14.

2. Three way choice: Exercise 12.15 or 12.17 or (12.11 and 12.13).

3. Two way choice: Exercise 15.13 or 15.15.

4. Consider the irreducible polynomial with integer coefficients,

   f(x) = (x - sqrt(2) - sqrt(3)) (x - sqrt(2) + sqrt(3)) 
          (x + sqrt(2) - sqrt(3)) (x + sqrt(2) + sqrt(3))
   

   The polynomial f(x) factors as a product of two quadratic factors in GF(p) for each of p = 5 and p = 7. In one of these factorizations the factors have no linear terms. Which one? Give a proof of your answer reasoned so as to involve as little arithmetic computation as possible.

5. Four way choice: 16.2 or 16.4 or 16.6 or 16.7.