Finding the Intersection of Subfields in Finite Fields

[mathmari]

Hey!

Let $\mathbb{F}_{p^m}$ and $\mathbb{F}_{p^n}$ be subfields of $\overline{\mathbb{Z}}_p$ with $p^n$ and $p^m$ elements respectively.

To find the field $\mathbb{F}_{p^m} \cap \mathbb{F}_{p^n}$ I have done the following:

Let $\mathbb{F}_{p^n} \cap \mathbb{F}_{p^m} = \mathbb{F}_{p^d}$.

$\mathbb{F}_{p^a} \leq \mathbb{F}_{p^b} \Leftrightarrow a \mid b$

We have that $\mathbb{F}_{p^d} \leq \mathbb{F}_{p^n} \Rightarrow d \mid n$ and $\mathbb{F}_{p^d} \leq \mathbb{F}_{p^m} \Rightarrow d \mid m$

So, $d \mid \gcd(n,m)$. We have that $\gcd(n, m) \mid m \Rightarrow \mathbb{F}_{p^{\gcd(n,m)}} \leq \mathbb{F}_{p^m}$ and $\gcd(n, m) \mid n \Rightarrow \mathbb{F}_{p^{\gcd(n,m)}} \leq \mathbb{F}_{p^n}$Does it stand that if $A \leq B$ and $A \leq C$, then $A\leq (B\cap C)$ ?? (Wondering)

If it stands, then we have that $\mathbb{F}_{p^{\gcd(n, m)}} \leq \mathbb{F}_{p^d} \Rightarrow \gcd(n, m) \mid d$



So, $d \mid \gcd(n, m)$ and $\gcd(n, m) \mid d$.

That means that $d=\gcd(n, m)$.

Therefore, $\mathbb{F}_{p^n} \cap \mathbb{F}_{p^m} = \mathbb{F}_{p^{\gcd(n, m)}}$

Is this correct?? (Wondering) Or could I improve something ?? (Wondering)