Tutorial 2 : Symmetric Polynomials and Splitting Fields

1. Let be a field and
where
Show that

disc

Note that

Therefore
Hence we have

disc

2. Show that
disc

The roots of the polynomial
are of the form
where

and

Hence

disc

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

(a) Let be a root of
The
Hence
Therefore using the formula for
disc in
terms of derivatives, we get

disc

(b) Let
be monic polynomials and
Let
be the roots of

disc

disc

disc

(c) Let
and
Then
Hence

discdiscdisc

This gives the desired formula.

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

If
then defines
an automorphism of given by
Let be fixed by all even permutations. Consider the polynomials
and
where is a transposition. We show that is symmetric and is
antisymmetric, i.e.
sign Let be an even permutation. Then

A similar argument shows that for odd we have
Hence is
symmetric.
Now let be an even permutation. Then

sign

A similar argument shows that for an odd permutation we have
sign Since vanishes if we put it follows that
Therefore
where
Since is
antisymmetric, is symmetric.

5. Find the splitting field of

Let
Check that is irreducible
over
by Eisenstein's criterion. Note that

Let
Then the roots of are
Hence
is a splitting field of over
Hence

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

Let
and
Then
If
is an -embedding
of fields where
are extensions of
then
for is called a conjugate of
Conjugates of are
Conjugates of
are
Hence
See the following figure

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

Let
.Then

Thus
are all the root of
Suppose has no root in and
where deg
and
The coefficient of in is sum of
elements in the set
Hence
it is
where But
since
This is contradiction.

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

Let
where
are all
roots of
in
Let be an algebraic closure of containing .
Then
is an embedding and
is algebraic.Thus can be extended to an embedding of into
But maps the set
to
Since contains a unique splitting field of over namely
Hence

9. Find a splitting field of over
Find

Let
where
are all
roots of
in
Let be an algebraic closure of containing .
Then
is an embedding and
is algebraic.Thus can be extended to an embedding of into
But maps the set
to
Since contains a unique splitting field of over namely
Hence