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The answer is no in general, as shown by the following example.

Let $f \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).

Then the ramification locus is empty. However, the generator $f$ of the $\mathbb Z / 2 $-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.

Indeed, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$.

The answer is no in general, as shown by the following example.

Let $\pi \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).

Then the ramification locus is empty. However, the generator $f$ of the $\mathbb Z / 2 $-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.

Indeed, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$.

The answer is no in general, as shown by the following example.

Let $f \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).

Then the ramification locus is empty. However, the generator $f$ of the $\mathbb Z / 2 \mathbb Z$-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.

Indeed, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$.

The answer is no in general, as shown by the following example.

Let $f \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).

Then the ramification locus is empty. However, the generator $f$ of the $\mathbb Z / 2 $-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.

Indeed, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$.

The answer is no in general.

For instance, let $f \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).

Then the ramification locus is empty. However, the generator $f$ of your $\mathbb Z / 2 \mathbb Z$-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.

In fact, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$.

The answer is no in general, as shown by the following example.

Let $f \colon X \to Y$ be an étale (i.e., unramified) double cover of projective curves over $\mathbb{C}$, with $g(X) \geq 3$ (for instance, we can take $g(X)=3$, $g(Y)=2$).

Then the ramification locus is empty. However, the generator $f$ of the $\mathbb Z / 2 \mathbb Z$-extension $K(X)/K(Y)$ must necessarily vanish somewhere on $X$.

Indeed, since $X$ is compact, there exist no non-constant holomorphic functions on it, hence $f$ has at least one pole. But the Residue Theorem implies that the number of zeroes of $f$ equals the number of poles (if counted with multiplicities), so $f$ has at least one zero on $X$.