How to show the degree of every factor of unique factorization of $f(x)$ over $L$ is 1?

Question

Let $L$ be a finite field extension of $K$ and $a\in L$. Suppose $a$ is algebraic over $K$, let $f(x)$ be the minimal polynomial of $a$ over $K$. How to show the degree of every factor of unique factorization of $f(x)$ over $L$ is 1?

Answer 1

As stated, this is not true. Let $K = \Bbb F_2(t)$ and $L=K(\sqrt{t})$, a degree two extension of $K$. Then the minimal polynomial of $\sqrt{t}$ over $K$ is $x^2+t^2 = (x+t)^2$. What you're really looking for is a separable extension of fields, where this follows more or less by definition.