## Tutorial 2 : Symmetric Polynomials and Splitting Fields

1. Let be a field and where Show that
disc

2. Show that disc

3. (a)Show that
(b)
Let and be monic polynomials and Show that
(c)
Show that

4. Show that a polynomial where is a commutative ring having is fixed under all the automorphisms of induced by even permutations in if and only if where and are symmetric polynomials and

5. Find the splitting field of

6. Let be a splitting field of over Find a complex number such that

7. Let be a field of characteristic Let Show that either all roots of lie in or is irreducible in

8. Let be a splitting field over a field of Let be a subfield of the field extension Let be an -embedding. Show that can be extended to an automorphism of

9. Find a splitting field of over Find