Both $x^{p-1} + p$ and the cyclotomic polynmial $\Phi_p(x) = (x^p-1)/(x-1)$ are irreducible over $\mathbf Q_p$ with degree $p-1$ and a root of either polynomial generates an extension of $\mathbf Q_p$ containing all the other roots of the polynomal, so it suffices to show the field generated over $\mathbf Q_p$ by a root of either polynomial contains at least one root of the other polynomial.
In Lemma 14.6 in the first edition of Washington's book "Introduction to Cyclotomic fields" (the chapter on the Kronecker-Weber theorem, and probably also Lemma 14.6 in the second edition), this is proved by showing that when $\zeta_p$ is a nontrivial $p$th root of unity, $(\zeta_p - 1)^{p-1}/(-p)$ can be written as a $(p-1)$th power in $\mathbf Q_p(\zeta_p)$, say $a^{p-1}$. Then $-p = ((\zeta_p - 1)/a)^{p-1}$, so $\mathbf Q_p(\zeta_p)$ contains a root of $x^{p-1} + p$. This is also in Gouvea's book "$p$-Adic Numbers: an Introduction": section 5.6 in the 1st edition and section 6.7 in the 3rd edition. It is not numbered as a theorem, but the section is only several pages long. The argument there is similar to what Washington does, but with more details given.
In the proposition (not numbered) in section 2 of Chapter 7 of Robert's "A first course in $p$-adic analysis", writing $\pi$ for a root of $x^{p-1} + p$, it's shown how to build a nontrivial $p$th root of unity in $\mathbf Q_p(\pi)$ using an infinite series: $|\pi|_p = (1/p)^{1/(p-1)}$, the formal power series $f(x) := e^{x + x^p/p}$ has radius of convergence bigger than $|\pi|_p$, and $f(\pi)$ is a nontrivial $p$th root of unity. This is surprising since naively you'd think $f(x)$ evaluated at a nonzero root of $x + x^p/p$ should have value $e^0 = 1$, but that's not what happens.
The reason our naive expectations are not met is that naively we think when $\pi + \pi^p/p = 0$ that $e^{\pi+\pi^p/p} = e^\pi{e}^{\pi^p/p} = e^\pi{e}^{-\pi} = e^0 = 1$, and the mistake there when $\pi \not= 0$ is in the 1st equation: $e^\pi$ and $e^{\pi^p/p}$ make no sense since $|\pi|_p = |\pi^p/p|_p = (1/p)^{1/(p-1)}$ and $e^x$ converges only when $|x|_p < (1/p)^{1/(p-1)}$. It is true (but not at all obvious) that $e^{x+x^p/p}$ as a formal power series converges at the $p$-adic roots of $x + x^p/p$, but the naive reason to think the value at those roots equals $1$ is wrong, and the value there really isn't $1$ (when $\pi \not= 0$).
In particular, $f(\pi)$ is not $e^{\pi + \pi^p/p}$. That is,
$e^{x+x^p/p}|_{x=\pi} \not= e^{\pi+\pi^p/p}$; evaluating a composition of formal power series $g(h(x))$ at a number $a$ is not actually defined to be $g(h(a))$; showing the evaluation really is $g(h(a))$ requires a proof, and sometimes the evaluation of $g(h(x))$ at $x = a$ simply isn't $g(h(a))$.
When we take $p$-th powers, $(e^{x+x^p/p})^p = e^{px + x^p}$ as formal power series, and when $\pi + \pi^p/p = 0$ we have $|p\pi|_p < (1/p)^{1/(p-1)}$ and $|\pi^p|_p = |-p\pi|_p < (1/p)^{1/(p-1)}$, so it is true that $e^{p\pi + \pi^p} = e^{p\pi}e^{\pi^p} = e^{p\pi}e^{-p\pi} = e^0 = 1$ even though it's not true that $\pi + \pi^p/p = 0 \Rightarrow e^{x+x^p/p}|_{x=\pi} = 1$. So $\zeta_{\pi} := e^{x+x^p/p}|_{x=\pi}$ is a nontrivial $p$th root of unity when $\pi$ is a nonzero root of $x + x^p/p$.
The second approach is more technical than the first approach (both to show $f(x)$ has radius of to show $f(\pi)^p = 1$ with $f(\pi) \not= 1$), so I'd advise citing Washington's book.
Something the second approach reveals is a nice bijection between the roots of $\Phi_p(x)$ (nontrivial $p$th roots of unity) and the roots of $x^{p-1} + p$. The different roots of $\Phi_p(x)$ are at distance $(1/p)^{1/(p-1)}$ from each other, and the same is true for the different roots of $x^{p-1} + p$. When $\alpha$ runs over roots of $\Phi_p(x)$ and $\beta$ runs over roots of $x^{p-1} + p$, by asking for
$$
|(\alpha - 1) - \beta|_p < \left(\frac{1}{p}\right)^{1/(p-1)}
$$ we get a unique $\beta$ for each $\alpha$ and a unique $\alpha$ for each $\beta$. To each $\beta$, the choice of $\alpha$ is $f(\beta) = e^{x+x^p/p}|_{x = \beta}$. And this also works between roots of $x^p-1 = (x-1)\Phi_p(x)$ and $x^p/p + x = (x^{p-1} + p)x/p$, where $\alpha = 1$ is matched with $\beta = 0$.
