Identify all the subfields of a field $\displaystyle F=F_{p^{18}}$, with $\displaystyle p^{18}$ elements where $\displaystyle p$ is a prime. Draw the lattice of all subfields.
I am allowed to just use a theorem, but I don't know one to use.
Identify all the subfields of a field $\displaystyle F=F_{p^{18}}$, with $\displaystyle p^{18}$ elements where $\displaystyle p$ is a prime. Draw the lattice of all subfields.
I am allowed to just use a theorem, but I don't know one to use.
The subgroups of $\displaystyle \mathbb{F}_{p^{18}}$ are $\displaystyle \mathbb{F}_{p^{k}}$ where $\displaystyle k|18$. Furthermore, $\displaystyle \mathbb{F}_{p^{k}}\subseteq \mathbb{F}_{p^{j}}$ if and only if $\displaystyle k|j$.
Now you can draw the diagram of all subfields.
Hint: The subfield diagram should look like an inverted $\displaystyle \mathbb{Z}_{18}$ diagram*.
(*This is not an accident! It follows from fundamental theorem of Galois theory.)
http://lh5.ggpht.com/_miqRA1rlzTU/ST...dpZsJ6Fs/1.jpg
Here it is. I am pretty sure it is correct though. Thanks for the help.