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For every non-trivial irreducible character $\chi$ of a finite $p$-group $P,$ we may choose an element $z in P$ such that $\chi(z) = \chi(1) \omega$ for some primitive $p$-th root of unity $\omega$ (this is an easy consequence of Schur's Lemma and the fact that the image of $P$ in the associated representation has non-trivial center). Now let $\zeta$ be a primitive $p^{e}$-th root of unity, where $P$ has exponent $p^{e}.$ Then ${\rm Gal}(\mathbb{Q}[\zeta]/\mathbb{Q}]$ acts on the irreducible characters of $P,$ and it is clear that $\chi$ is in an orbit of length divisible by $p-1.$ Since $\chi$ was an arbitrary non-trivial irreducible character, and since all irreducible characters in the same orbit have the same degree it follows both that the total number of non-trivial irreducible characters, and the total number of irreducible non-linear characters of $P$ are multiples of $p-1$ (in fact, the number of irreducible characters of $p$ of any fixed degree $p^{d} >1$ is a multiple of $p-1.$