Let G be any field
a)Prove that for any primary $q, G$ contains at most one subfield isomorphic to $\mathbb{F}_q$
b) If $\mathbb{F}_{p^m}$ and $\mathbb{F}_{p^n}$ are subfields of $G$, prove that $\mathbb{F}_{p^m}\bigcap \mathbb{F}_{p^n} = \mathbb{F}_{p^d}$ where $d=gcd(m,n)$
Here are my attempts:
a) I know that a field $\mathbb{F}_{p^n}$ has a subfield isomorphic to $\mathbb{F}_{p^m}$ iff $m|n$ so in the case of $G$ being finite if $G=\mathbb{F}_{q^n}$ it will have a subfield isomorphic to $\mathbb{F}_q$. If $G$ is finite and of another form it will not contain a subfield of $\mathbb{F}_q$. Not sure about what to do if $G$ is infinite (if we even need to look at that is unclear since the name of this section is finite fields) or if this is even right
b) $\mathbb{F}_{p^n}$ and $\mathbb{F}_{p^m}$ are subfields of $G$ so that means $G = \mathbb{F}_{p^k}$ for some $k$ and $m|k$ and $n|k$ so $gcd(m,n)|k$. Not sure where to go from here
For a), it looks like you're trying to prove existence whereas the question only asks for uniqueness assuming existence. As a hint, consider the polynomial $f=x^q-x\in\mathbb{F}_q[x]$.
For b), you're assuming that $G$ is finite, which is unnecessary. Instead, notice that $\mathbb{F}_{p^m}\cap\mathbb{F}_{p^n}$ will be a subfield of both $\mathbb{F}_{p^m}$ and $\mathbb{F}_{p^n}$. Try to use the fact you stated about existence of subfields of finite fields and what you proved in a) to show that $|L|=gcd(m,n)$.