Prove that the $F$-algebra $K_1 \otimes K_2$ is a field iff $[K_1K_2:F] = [K_1:F][K_2:F]$

Question

I am trying to prove that the $F$-algebra $K_1 \otimes K_2$ is a field iff $[K_1K_2:F] = [K_1:F][K_2:F]$ where $K_1$ and $K_2$ are finite extensions of a field $F$ contained in the field $K$. This is a problem coming from Dummit and Foote in chapter 13.2 page 531. After trying to attack the problem for hours, I tried finding some answers online. In one of the solutions I found, the authro claims that the dimension of $K_1 \otimes_F K_2$ as a free module over $F$ is equal to the degree of the extension $K_1 \otimes_F K_2 /F$. I am struggling to see this statement. Why is this true? Is this just a fact from module theory or can we prove this? If so, can anyone give me a hint for how to prove it? Thanks!

Answer 1

The statment you asked about is true by the definition of the degree of an extension.