# Thread: subfields of a field

1. ## subfields of a field

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.

2. Originally Posted by numbertheory12
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.)

3. I got this (I don't know how draw lattice diagrams in latex): Is that correct?

4. Originally Posted by numbertheory12
I got this (I don't know how draw lattice diagrams in latex): Is that correct?
I do not see any picture.
You can put it into Paint and then upload it to MHF.

5. http://lh5.ggpht.com/_miqRA1rlzTU/ST...dpZsJ6Fs/1.jpg

Here it is. I am pretty sure it is correct though. Thanks for the help.

6. Good job. You just missed one more field, $\displaystyle \mathbb{F}_p$, the base field.
It goes below $\displaystyle \mathbb{F}_{p^3}$ and $\displaystyle \mathbb{F}_{p^2}$.