Factoring $x^n-1$

Recall that every element $\beta \in GF(q^m)$ has an order that divides $q^m-1$; thus every element is a root of $x^{q^m-1}-1$. Put another way, the elements of $GF(q^m)$ are the $(q^m-1)$st roots of unity, and these are all the nonzero elements of the field.

Given a field, we divide it into conjugacy classes, taking the minimal polynomial from each. Then based on our observation, we must have $x^{q^m-1}$ as a product of the minimal polynomials of the nonzero elements.

We can now pursue a more general problem, roots of $x^n-1$ for other values of $n$. These $n$ roots of unity must exist in some field. We find the field, then find the minimal polynomials of the conjugacy classes in the field.

Suppose we have an element $\beta$ of order $n$ in some field $GF(p^m)$. Then $\beta$ is a root of $x^n-1$ in that field, and so are the elements $\beta^2, \beta^3, \ldots, \beta^{n-1}$. We can find such a $\beta$ if we form the field correctly.

Before doing so, it is interesting to pause a moment and suggest an application of this. Consider the DFT:

$$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi nk/N}.$$

Note that the number $e^{-j2\pi/N}$ is an $N$th root of unity. We can define a Fourier transform in a finite field $GF(p^m)$ of length $N$, provided that we can find a field in which $N$th roots of unity exist.

Recall that if $n\vert p^m-1$, then there are $\phi(n)$ elements of order $n$ in $GF(p^m)$.

If $m$ is the order of $q$ modulo $n$, then $GF(q^m)$ is the smallest extension field of $GF(q)$ in which $n$th roots of unity exist.







If we list the exponents of the primitive roots of unity, we get what are called the cylotomic cosets. For example, for the last example we have the following:

Conjugacy class		Cyclotomic cosets
$\{1 \}$	$\leftrightarrow$	{0 }
$\{ \gamma, \gamma^7, \gamma^{4}, \gamma^{13} \}$	$\leftrightarrow$	{ 1,7,4,13}
$\{ \gamma^2, \gamma^{14}, \gamma^{8}, \gamma^{11} \}$	$\leftrightarrow$	{2,14,8,11}
$\{ \gamma^{3}, \gamma^{6}, \gamma^{12}, \gamma^{9} \}$	$\leftrightarrow$	{3,6,12,9}
$\{ \gamma^5 \}$	$\leftrightarrow$	{5}
$\{ \gamma^{10} \}$	$\leftrightarrow$	{10}

The cyclotomic cosets modulo $n$ with respect to $GF(q)$ contain the exponents of the $n$ distinct powers of a primitive $n$th root of unity with respect to $GF(q)$, each coset corresponding to a conjugacy class. These cosets provide a shorthand representation for the conjugacy class.