Suppose we work in the number field $\mathbb Q(\omega_{pq})$ with $\omega_{pq}$ a $p q$th root of unity, where $p$, $q$ are distinct primes. Similarly $\omega_{p}$ and $\omega_{q}$ are $p$th and $q$th roots of unit respectively. Now we note the binomial expansion
$$
(1-(1-\omega_{p}))^{\frac{p}{q}}=1-\frac{p}{q}(1-\omega_{p})+ \binom{p/q}2(1-\omega_{p})^{2} + \dotsb.
$$


By simple analysis, we know it should converge in the non-archimedean topology to one of the $q$th roots of unity $\omega_{q}^{k}$ at a certain place $\beta$ over $1-\omega_{p}$, or equivalently over $p$. Denote the limit by $s$. Again by simple analysis of the expansion of $s-1$, we know $s\neq1$.



So we know $1, s, s^{2}, \dotsc, s^{q-1}$ will go once through all the $q$th roots of unity while all of them belong to $1+\pi\mathbb Z_{\beta}$ where $\pi$ is a primal element of place $\beta$. So $1+s+s^{2}+\dotsb+s^{q-1}\in q+\pi\mathbb Z_{\beta}$, so $\neq0$, which is a contradiction of the property of a $q$th root of unity.

What is wrong in the above argument?