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# Tag Archives: Abstract Algebra

## On the centralizer

Suppose that has this property that if is an eigenvalue of then is not an eigenvalue of . Show that if and only if for any . In other words the centralizer of equals the centralizer of .

**Solution**

It is clear that implies for any . Now suppose that for some and set . We want to prove that . We have

and so . It now follows that for any integer and thus for any and any integer we have

where is the identity matrix. Now let be a generalized eigenvector corresponding to an eigenvalue of . Then for some integer and thus, by we have . Therefore, since we are assuming that is not an eigenvalue of , we must have . So, since every element of is a linear combination of some generalized eigenvectors of , we get for all , i.e. and hence .

The exercise can also be found here.

## Isomorphic groups

Let . Define the group

Prove that where is the dihedral group.

**Solution**

Using or equivalently we can write each element of in the form where . Using we may assume that . Using we may also assume that . It is easy to prove inductively that .

Let . We prove that . Obviously . Furthermore, since

If (such that ) then hence and therefore or . If then hence which is a contradiction since .

Therefore,

which is precisely the dihedral group with elements.

## Inequality on groups

Let be a finite group and suppose that , are two subgroups of such that and . Show that

**Solution**

Recall that and thus . Hence and so

(1)

where and .

Now, since and we have and that is and . So if we let and then and thus

due to .

The exercise along its solution have been migrated from here .

## Irreducible factors of a polynomial

Let and let

Find all irreducible factors of .

**Solution**

Setting we note that

Hence

(1)

It’s clear that is irreducible over . Now, for let be the -th cyclotomic polynomial. Using well-known properties of , we have

Thus is irreducible over because cyclotomic polynomials are irreducible over . Hence, by ( 1 ) has exactly irreducible factors and they are .

## Galois theory … of the Euler’s totient function

Let and let be an -th primitive root of unity. Prove that

where denotes the Euler’s totient function.

**Solution**

We have

where since is the minimal polynomial of over .

The result follows.