Question

If F be a subfield of a finite field K. If F has order q show that K has order qⁿ where $\mathbf{n}\mathbf{=}\left[K:F\right]$ .

Short answer

It is proved that K has order qⁿ.

Step-by-step solution

Definition of subfield

A subfield of a mathematical field that is itself a field that mean a subset of a given field which is a field in itself is called as a subfield.

Showing that K has order qn  

Let F be a subfield of finite field K and F has order q . Let the characteristic of F be p.

That is:

$\begin{array}{rcl}\mathrm{charF}& =& \mathrm{p}\\ \mathrm{charF}& =& \mathrm{charK}\\ \mathrm{charK}& =& \mathrm{p}\\ \mathrm{charF}& =& \mathrm{charK}=\mathrm{p}\end{array}$

According to theorem given below:

A finite field F has order pⁿ where p is the characteristic of F and $m=\left[F:Z_p\right]$. Here F has order q.

Therefore $q=p^n$.

Now since:

$\begin{array}{rcl}\left[\mathrm{K}:{\mathrm{Z}}_{\mathrm{p}}\right]& =& \left[\mathrm{K}:\mathrm{F}\right]\left[\mathrm{F}:{\mathrm{Z}}_{\mathrm{p}}\right]\\ & =& \mathrm{mn}\\ {\mathrm{q}}^{\mathrm{m}}& =& {\mathrm{p}}^{\mathrm{n}}\end{array}$

Hence the order of K is qⁿ.