1. Let

be an extension of degree

** (a)** For any

prove that the map

defined by

for all

is a linear transformation of
the

-vector space

Show that

is isomorphic to a subfield
of the ring

of

matrices with
entries in

** (b)** Prove that

is a root of the characteristic polynomial of

Use this procedure to find monic polynomials satisfied by

and

2. Prove that

is not a sum of squares in the field

where

3. Let

be a field and

be an indeterminate.
Find the irreducible polynomial of

over

where

Clearly

is a root of

Since

is transcendental over

,

is an irreducible element of

Therefore the polynomial

is irreducible by Eisenstein criterion. Hence

irr
4. Find an algebraic extension

of

such that the polynomial

has a root in

Let

be a root of

Then
we have

and hence

But then deg of L.H.S. is even and deg of R.H.S. is odd.Thus

is
irreducible over

Hence

is a maximal ideal in

Hence

is an algebraic extension of

containing a root

of

5. The construction of a regular

-gon amounts to the construction
of the real number

Show that

is a root of

Hence conclude that a regular

-gon is not
constructible by ruler and compass.

Consider the diagram of fields:

Since

irr we have

Hence

irr As

Let

The

satisfies the equation

.
Hence

.
Thus

is not constructible by ruler and compass.

6. Show that an angle of

degrees,

is constructible if and only if

First we show that

is not constructible. Suppose it is,then a regular

-gon is constructible.But

.Hence

is not
constructible. Since

a

-gon is constructible.
Thus

is constructible. Suppose that

is constructible and

then

for some

.
Hence

is constructible.This is a
contradiction. Hence

. Conversely,suppose

, then

and
since

is
constructible,

is constructible.

7. Prove that it is impossible, in general, to quintsect an
arbitrary angle by ruler an compass. Is it possible to divide
the angle

degrees into five equal parts by ruler and compass ?

It is not possible to quintsect an angle in general.
Suppose that we could quintsect

Then an angle of

is constructible. Hence we can construct an angle of one degree. This is
a contradiction. We can construct

since

and
hence

can
be quintsected.