$K/k$ field extension, $[K:k]=p$ prime, $f(X)\in [X]$ has degree $p+1$, $f$ has root in $K\iff f$ has root in $k$

Question

Let $K/k$ a field extension, $[K:k]=p$ prime, $f(X)\in [X]$ with degree $\deg(f)= p+1$, so $f$ has root in $K\iff f$ has root in $k$.

As $k\subset K$, the $\Leftarrow$ implication is trivial. Any hints to the $\Rightarrow$ implication? I really don't know what to do.

Answer 1

Here's one Hint:

If $\xi\in K$ is a root of $f$, consider the subextension $$k\subset k(\xi)\subset K$$ and remember the degree of extensions is multiplicative.