Final exam

The grades have been published.

The grades are decided by following central guidelines that apply for mathematics courses. Each grade also reflects the individual exam solution.

Problem 1:

1a: 1-2 points reduction for each of incomplete or incorrect: associativity, closure of operation, existence of inverse in G, non-abelianity of G, order of G.

1b: 1-2 points reduction for each of incomplete or incorrect: elements of order 3, existence of normal subgroup.

Problem 2:

2a: maximum number of points requires correct and complete proof that G is a union of its elements of order d.

2b: maximum number of points requires correct and complete proof based on estimating the number of elements of order d using the given hypothesis and applying part 2a to get the asserted equality for all divisors, then specialising to n.

Problem 3:

3a: maximum number of points requires explanation that factors in x^4 +1 cannot belong to the ideal

3b: up to 2 points reduction for each of incomplete or incorrect: why a zero of f(x) is a generator for F_16* (either using ?feedback shift register? or Lagranges theorem); why f(x) does not have linear factors or factors of degree 2.

3c: maximum number of points requires using the known fact that a field with 16 elements is unique up to isomorphism  

3d: up to 2 points reduction for incomplete or incorrect explanation of which powers of theta appear in the factorisation

Problem 4:

4a: up to 4 points reduction for incomplete or incorrect deduction of the degree of extension

4b: up to 3 points reduction for each of incomplete or incorrect: deduction of the Galois group; deduction of the fixed fields associated to subgroup of the Galois group.