Intersection of Simple Extensions

Question

Let $F/K$ be field extension. Let $a,b\in F$. Show that if $[K(a):K]=m$, $[K(b):K]=n$ and $gcd(m,n)=1$, then $K(a)\cap K(b)\subseteq K.$

Also, is it true that elements of K(a) are of the form {$x_1+x_2a+x_3a^2+...+x_ma^{m-1}$| $x_i\in K$}

This question is small part of my homework question and I would be happy if I understand the notion of elements of simple extension. Can someone help me with the answer?

Answer 1

One has $K(a)\cap K(b)\subset K(a)$ and $K(a)\cap K(b)\subset K(b)$. Therefore

$$m=[K(a):K(a)\cap K(b)][K(a)\cap K(b):K]$$

and

$$n=[K(b):K(a)\cap K(b)][K(a)\cap K(b):K]$$

So we have proven that $[K(a)\cap K(b):K]$ divides $m$ and $n$ and so

$$[K(a)\cap K(b):K]=1$$

This means $K(a)\cap K(b)=K$