The genus of a field over $\mathbb C$ is invariant under isomorphisms of $\mathbb C$-algebras

For a field $K$ containing $\mathbb C$, let $\Omega_{K/\mathbb C}$ be its module of Kähler differentials, with universal derivation $d$. A place of $K$ is a valuation ring $\mathcal O\ne K$ of $K$ (a subring with $\xi\in\mathcal O$ or $\xi^{-1}\in\mathcal O$ for every nonzero $\xi\in K$) that contains $\mathbb C$. A differential is regular at $\mathcal O$ if it is a $\mathbb C$-linear combination of differentials $a\,db$ with $a,b\in\mathcal O$. The holomorphic differentials $H(K)\subseteq\Omega_{K/\mathbb C}$ are those regular at every place of $K$, and the genus of $K$ is $g(K)=\dim_{\mathbb C}H(K)$.

Let $K$ and $L$ be fields containing $\mathbb C$, and suppose there is an isomorphism of $\mathbb C$-algebras $K\cong L$. Then

So the genus depends only on the field, not on a plane model. It transfers the genus computed on the double cover of the family to the function field of the curve $P_\alpha=X^m(\alpha+XY)+Y^m(\bar\alpha+XY)=0$ (Lemma 4), and in Remark 5 it is applied to the isomorphism of function fields that a similarity between $\operatorname{Re}(z^4)=1$ and a curve of the $m=2$ family would induce, the two genera being three and two.

Formalization Note: $g(K)$ is Module.finrank, which is $0$ for an infinite-dimensional space; the isomorphism is a term of K ≃ₐ[ℂ] L.