We know that quadratic subfield of

is

.
Hence

contains

and

contains

. If

,then

and

are subfields of

. However

contains a
unique quadratic subfield. Hence

and therefore

by the
diagram below.

2. Determine a primitive element of a subfield

of

so that

The Galois group

is a cyclic group of order

Hence it has a unique
subgroup of order

for each divisor

of

. Hence there is a
unique subfield

with

The powers of

mod

are given in the table below.

Hence

Put

The automorphism

generates the
Galois group

Then

is a
subgroup of

having order

Put

Then

and

Thus

as there is a unique
intermediate subfield of degree

over

for every divisor

of

Hence a primitive element of

is

The powers of

modulo

are :

and

Hence

and the automorphism defined by

is a generator of the
Galois group

. Thus to find the degree of

,
we find the orbit of

under the action of

Thus there are

conjugates of

and hence

. The
orbit of

under the action
of

is :

Hence

.

The powers of

modulo

are:

Therefore

is a generator for

. The
orbit of

is

Thus

has only two conjugates and therefore

5. Let

be a primitive element modulo

where

is a prime.
Thus

Let

Using the list

show how to find the sum

of powers of

which determines
a subfield

of

so that

where

The map

is a generator of

Then

is the unique subgroup of

of order

Let

Then

Hence a primitive element of

is

6. Suppose

and

for some integer

Show that

can be diagonalized. Prove that the
matrix

where

and

is a field of chracteristic

satisfies

and cannot be diagonalized if

By the equation

we see that the
minimal polynomial of

is a divisor of

which has distinct roots.
Hence

can be diagonalized.

The minimal polynomial of

divides

which has only one root namely
1. Hence if there exists

so that

This is a contradiction.

Consider the fractions

Reduce them to lowest terms. Then the fraction with denominator

where

will be

in number. The total
number of fractions is

. Thus

.By
M

bius
inversion

.

Use the formula

and apply the multiplicative version of the M

bious inversion
formula.

9. Show that

and deduce that
the coefficients of

satisfy

for all

We know that

is a root of

if and only if

is a root of

Verify that the product of the roots of

is

Note that if

is a root of

then

This translates into the

*palindromic* property of