= Solution
A <function element> is a pair $(f,D)$ where $D$ is a nonempty disk and $f$ is holomorphic there; the germ at its center captures its local information. Elements continue one another when their <functions> agree on an overlapping <connected> neighborhood. A <complete analytic function> is a maximal collection reachable from one element by chains of <analytic continuation>, including all further continuations of its members.
If $f^n=p$ in the initial disk, the <identity theorem> propagates this identity across every overlapping disk in a continuation chain. Induction along the chain proves \b[every element of the <complete analytic function> also satisfies $\widetilde f^{\,n}=p$].
Put $F(z,w)=w^n-p(z)$. Its gradient is $(-p'(z),nw^{n-1})$. At a point of the curve with $w\ne0$, the second entry is nonzero; at $w=0$, the root of $p$ is simple and the first entry is nonzero. Thus the <holomorphic implicit function theorem> supplies charts. Its precise local statement is that a holomorphic equation with nonzero derivative in one variable can locally be solved uniquely and holomorphically for that variable as a function of the other. For $w\ne0$, use $z$ as coordinate and write $w=w(z)$; at a simple root use $w$ as coordinate and write $z=z(w)$. The transition maps are holomorphic with nonzero derivatives, giving a <Riemann surface>. The two <projections> are holomorphic because they are holomorphic expressions in either kind of chart.
Away from the zeros of $p$, the projection onto $z$ is an unramified $n$-sheeted covering. A germ of a root corresponds to $(z,f(z))$ and its analytic continuations follow this covering. Near a simple root, $z-z_0$ is a nonzero analytic factor times $w^n$; a circuit about $z_0$ cyclically permutes all $n$ branches. For nonconstant square-free $p$, these circuits connect all sheets. Hence the germ surface $S(F)$ is the curve with the points over the roots removed, and \b[$C$ is its branched completion obtained by adding those ramification points]. An ordinary function element in the variable $z$ cannot include such a point: the zero order of a holomorphic $n$th power is divisible by $n$, whereas $p$ has zero order one. If a convention already includes these branch points in the <Riemann surface> of a multivalued function, that completed surface is $C$ itself. For a nonzero constant $p$, the curve has $n$ separate components, and the chosen element determines just one of them. The identically zero polynomial is excluded by the non-singularity premise.
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