I am reading an algebra book by Fumiyuki Terada.
There is the following problem in this book:
$E_1, E_2, K$ are fields.
$K$ is a subfield of $E_1$.
$K$ is a subfield of $E_2$.
$p, q$ are prime numbers.
$[E_1 : K] = p, [E_2 : K] = q$.Then, $E_1 = E_2$ or $E_1 \cap E_2 = K$.
And Terada's answer to this problem is here:
Let be $B = E_1 \cap E_2$.
Then, $K \subset B \subset E_1$.
Therefore, $[E_1 : B] [B : K] = p$.
Because $p$ is a prime number, $[E_1 : B] = p, [B : K] = 1$ or $[E_1 : B] = 1, [B : K] = p$.
If $[E_1 : B] = p, [B : K] = 1$, then $B = K$.
If $[E_1 : B] = 1, [B : K] = p$, then $E_1 = B = E_1 \cap E_2$. $\therefore K \subset E_1 \subset E_2$. $[E_2 : E_1] [E_1 : K] = q$. $\therefore [E_2 : E_1] p = q$. Because $p, q$ are prime numbers, $p = q, [E_2 : E_1] = 1$. $E_2 = E_1$.
Why is $E_1 \cap E_2$ a field?
If there is a field $E$ such that $E_1 \subset E$ and $E_2 \subset E$ and $E_1, E_2$ are subfields of $E$, then I think it is easy to show that $E_1 \cap E_2$ is a field.
If $a, b \in E_1 \cap E_2$, then $a *_1 b \in E_1$ and $a *_2 b \in E_2$.
$a *_1 b = a *_2 b$?
I cannot understand this problem.