Let $L:K$ be a field extension and $a,b \in L$. Suppose $[K(a):K]=m$ and $[K(b):K]=n$
why is it that $[K(a,b):K]=mn$?
I’ve tried to solve this using the tower Law but to use this I need to show $[K(a,b):K(a)]=m$.
How is this certainly the case? Must the minimal polynomial of $b$ over $K$ and $K(a)$ necessarily be the same?
Here is a counterexample. Let $L=\mathbb{Q}(2^{1/4})$ and $K=\mathbb{Q}$. Then, if we $a=2^{1/2}$ and $b=2^{1/4}$ (and both belongs to $L$), then $$[\mathbb{Q}(2^{1/4}): \mathbb{Q}]=4,\qquad [\mathbb{Q}(2^{1/2}): \mathbb{Q}]=2$$ but $$[\mathbb{Q}(2^{1/4}, 2^{1/2}): \mathbb{Q}]=[\mathbb{Q}(2^{1/4}): \mathbb{Q}] =4 \neq 2\cdot 4 =8.$$ However, your statement holds when $m$ and $n$ are coprimes. You can see the proof here:
Product of degree of two field extensions of prime degree