Does adjoining an element to a field change the characteristic? (No it doesn't.)

Question

This may have an answer somewhere already, but I can't find it. Let $K$ be a field of characteristic $p$. Let's adjoin, for example, $\zeta_n$ where $\zeta_n$ is some primitive $n$-th root of unity. Assume $\zeta_n\not\in K$. Is it still the case that $K(\zeta_n)$ is of characteristic $p$? Does this hold in general, i.e. does adjoining an element ever change the characteristic of a field?