abstract algebra - Degree of field extension using minimum polynomial

Let $K \subset L$ an algebraic extension. I want to prove for $a,b \in L$ that

$$[K(b): K] \geq [K(a,b): K(a)],$$
Where for example $[K(b):K]$ is the degree of the field extension $K \subset K(b)$. My ideas so far are:

1. The degree of $[K(a):K]$ is equal to $\deg(f_K^a)$, where $f_K^a$ is the minimum polynomial. So
   $$K[X]/(f_K^a) \stackrel{\sim}{\to} K(a).$$
   and I want to find a minimum polynomial with root $a$.

My question is: how do I find the minimum polynomials?

Answer

You don't need to find the minimal polynomial. That's because whatever the polynomial is for $b$ over $K$, call it $p$, it is still a polynomial for $b$ over $K(a)$. Therefore, the minimal polynomial for $b$ over $K(a)$ divides $p$ and therefore has degree no greater than $\deg(p)=[K(b):K]$.