Ramification indices of the power map on the projective line

The morphism. The coordinate $x$ satisfies ${\mathrm{ord}}_{\mathrm{\infty }}(x)=-1$ by [F1], so $x$ is nonconstant and hence $x^n$ is nonconstant as well. By [F2] applied to $f=x^n$ there is a finite locally free morphism $\phi \text{\hspace{0.17em}⁣}:{\mathbb{P}}_k^1\to {\mathbb{P}}_k^1$ of degree $[k({\mathbb{P}}_k^1):k(x^n)]$ with ${\phi }^{\mathrm{♯}}(x)=x^n$; it is nonconstant, hence surjective. On the affine charts the comorphism is $k[x]\to k[x]$, $x\mapsto x^n$ on $U_0$, and $k[y]\to k[y]$, $y=x^{-1}\mapsto (x^n{)}^{-1}=y^n$ on $U_{\mathrm{\infty }}$. In the homogeneous coordinates of [F1] the target coordinate of the image of $[s:t]$ with $s\mathrm{\ne }0$ is $x^n=(t\mathrm{/}s{)}^n$, so the image is $[1:t^n\mathrm{/}{s}^n]=[s^n:t^n]$; thus $\phi$ is the morphism of the statement.

Zeros and poles. The order function of [F3] is additive, so ${\mathrm{ord}}_p(x^n)=n{\mathrm{ord}}_p(x)$ for every closed point $p$; by [F1] the only points with ${\mathrm{ord}}_p(x)\mathrm{\ne }0$ are $0$, where ${\mathrm{ord}}_0(x)=1$, and $\mathrm{\infty }$, where ${\mathrm{ord}}_{\mathrm{\infty }}(x)=-1$. Hence $\mathrm{div}(x^n)=n[0]-n[\mathrm{\infty }]$, the zero divisor of $x^n$ is $(x^n{)}_0=n[0]$, and its pole divisor is $(x^n{)}_{\mathrm{\infty }}=n[\mathrm{\infty }]$. By [F2] the fibre of $\phi$ over $0$ is carried by $n[0]$ and the fibre over $\mathrm{\infty }$ by $n[\mathrm{\infty }]$; in particular ${\phi }^{-1}(0)=\{0\}$ and ${\phi }^{-1}(\mathrm{\infty })=\{\mathrm{\infty }\}$ as sets, with $\kappa (0)=\kappa (\mathrm{\infty })=k$ by [F1].

Ramification at $0$ and at $\mathrm{\infty }$. By [F1] the element $x$ is a uniformizer of ${\mathcal{O}}_{{\mathbb{P}}_k^1,0}$, and the pullback of the target uniformizer $x$ at $0$ is ${\phi }^{\mathrm{♯}}(x)=x^n$, so $e_0={\mathrm{ord}}_0(x^n)=n$ by [F3]. At infinity $y=x^{-1}$ is a uniformizer of ${\mathcal{O}}_{{\mathbb{P}}_k^1,\mathrm{\infty }}$ by [F1], and the pullback of the target uniformizer $y$ at $\mathrm{\infty }$ is ${\phi }^{\mathrm{♯}}(y)=y^n$, so $e_{\mathrm{\infty }}={\mathrm{ord}}_{\mathrm{\infty }}(y^n)=n$.

The degree is $n$. The function field extension $k({\mathbb{P}}_k^1)\mathrm{/}k(x^n)$ is separable because $x$ satisfies the polynomial $T^n-x^n\in k(x^n)[T]$ whose derivative $n{T}^{n-1}$ has no common root with it in characteristic not dividing $n$, so [F4] applies to the nonconstant morphism $\phi$ of degree $[k({\mathbb{P}}_k^1):k(x^n)]$. Evaluating the fibre-degree sum of [F4] at $q=0$ and using that the fibre is the single point $0$ with $\kappa (0)=k$ by step 1.2 and $e_0=n$ by step 1.3 gives $\mathrm{deg}(\phi )=e_0[\kappa (0):k]=n$. The same computation at $\mathrm{\infty }$ gives $\mathrm{deg}(\phi )=e_{\mathrm{\infty }}[\kappa (\mathrm{\infty }):k]=n$, and the two readings agree.

Relative differentials on the two charts. On $U_0$ the map of affine charts is the ring map $A=k[x]\to B=k[x]$, $x\mapsto x^n$, so $B=A[T]\mathrm{/}(T^n-x)$ with $T\mapsto x$ and $g(T)=T^n-x$; its derivative is $g^{\mathrm{\prime }}(T)=n{T}^{n-1}$, and the class $n\in k$ is a unit because $\mathrm{char}k\nmid n$, so [F6] gives ${\mathrm{\Omega }}_{B\mathrm{/}A}\cong B\mathrm{/}(T^{n-1})=k[x]\mathrm{/}(x^{n-1})$. Localizing at the maximal ideal $(h)$ of a closed point $p=V(h)\in U_0$ as in [F1] and using the localization clause of [F6], the stalk is ${\mathrm{\Omega }}_{{\mathbb{P}}_k^1\mathrm{/}{\mathbb{P}}_k^1,p}\cong k[x{]}_{(h)}\mathrm{/}(x^{n-1})$: this is zero when $h\mathrm{\ne }x$, because then $x$ is a unit of the localization, and for $p=0$ it is the module $k[x{]}_{(x)}\mathrm{/}(x^{n-1})$, whose filtration by the powers of the uniformizer $x$ has $n-1$ successive quotients isomorphic to $\kappa (0)=k$, so its length is $n-1$ by [F7]. The same computation in the coordinate $y$ on $U_{\mathrm{\infty }}$ gives ${\mathrm{\Omega }}_{{\mathbb{P}}_k^1\mathrm{/}{\mathbb{P}}_k^1,p}=0$ for $p\in U_{\mathrm{\infty }}$ with $p\mathrm{\ne }\mathrm{\infty }$ and length $n-1$ at $\mathrm{\infty }$. Hence the relative differentials are supported exactly on $\{0,\mathrm{\infty }\}$, with $l_0=l_{\mathrm{\infty }}=n-1$, and this support is empty exactly when $n=1$.

Unramifiedness away from $0$ and $\mathrm{\infty }$. Let $p\mathrm{\notin }\{0,\mathrm{\infty }\}$ be a closed point. By step 2.2 the stalk ${\mathrm{\Omega }}_{{\mathbb{P}}_k^1\mathrm{/}{\mathbb{P}}_k^1,p}$ vanishes, so $l_p=0$, and the criterion of [F5] gives $e_p=1$ together with separability of $\kappa (p)\mathrm{/}\kappa (\phi (p))$; in the terminology of [F5] the point $p$ is unramified and lies in neither the index-ramification locus nor the differential-ramification locus. Since $e_0=e_{\mathrm{\infty }}=n$ by step 1.3, the index-ramification locus equals $\{0,\mathrm{\infty }\}$ when $n\ge 2$ and is empty when $n=1$, and by step 2.2 the same holds for the differential-ramification locus.

The ramification divisor. Define $R_{\phi }={\sum }_p{l}_p[p]$ with $l_p$ the length of the relative differentials at $p$; since $l_p=0$ for all but finitely many $p$ and all $l_p\ge 0$, this is an effective divisor on ${\mathbb{P}}_k^1$ in the sense of [F8]. By step 2.2 its coefficients are $l_0=l_{\mathrm{\infty }}=n-1$ and $l_p=0$ for every other closed point, so $R_{\phi }=(n-1)([0]+[\mathrm{\infty }])$, with support $\{0,\mathrm{\infty }\}$ and length $n-1$ at each of the two points when $n\ge 2$, and $R_{\phi }$ is the zero divisor when $n=1$.

Conclusion. The morphism $\phi ([s:t])=[s^n:t^n]$ of the statement is finite and surjective of degree $n$ by steps 1.1 and 2.1; it is unramified at every closed point away from $0$ and $\mathrm{\infty }$ by step 3.1; at $0$ and at $\mathrm{\infty }$ the fibre is the single point with ramification index $n$ by steps 1.2 and 1.3, so the fibre degree sum reads $n=n$ over both points by step 2.1; and, in the tame case $\mathrm{char}k\nmid n$, the ramification divisor is $R_{\phi }=(n-1)([0]+[\mathrm{\infty }])$ by step 3.2. The Axiom of Choice of [F9] is used only through the construction of the finite morphism and through the divisor theory of the projective line, and no further selection is made.