Is the formula $[k(a,b):k][k(a) \cap k(b):k] = [k(a):k][k(b):k]$ true for simple algebraic extensions ($k(a)/k$ and $k(b)/k$ )?

Question

Let $k$ be any field and $L/k$ be a field extension. Suppose $a, b \in L$ are algebraic over $k$. Is the formula $[k(a,b):k][k(a) \cap k(b):k] = [k(a):k][k(b):k]$ true?

This formula comes from page 228 (Remark 2.4) in "An invitation to Arithmetic Geometry" by Prof. Dino Lorenzini. But the author didn't explain why this formula is true. Thank you for four answer.

Answer 1

This is false. Consider $k=\Bbb{Q}$, $a=\root3\of2$, $b=e^{2\pi i/3}a$. We have $[k(a,b):k]=6$, $k(a)\cap k(b)=k$ and $[k(a):k]=[k(b):k]=3$.