the roots of $T^n -a$ in a splitting field.

Question

Let $K$ be a field and $a \in K$, $a \neq 0$. According to a text I'm reading, the roots of $T^n - a$ in a splitting field over $K$ are numbers of the form $\zeta \sqrt[n]{a}$, with $\zeta$ an nth root of unity in $K$. The proof is simple: Choose a fixed root $\alpha = \sqrt[n]{a}$ in the splitting field and if $\beta$ is another one, then $\alpha ^n = a = \beta ^n $, so $(\dfrac {\beta}{\alpha})^n = 1$. Set $\zeta = \dfrac {\beta}{\alpha} \in K$ etc. etc. However there is no explanation as to why $\zeta \in K$ and I really can't think of one. What am I overlooking? Or are they typing errors in the text?