Every finitely generated $\mathbb{C}$-algebra is finitely generated $\mathbb{Q}$-algebra.

Question

I want to know if every finitely generated $\mathbb{C}$-algebra is finitely generated $\mathbb{Q}$-algebra or not.

Although by Artin-Tate lemma we can see that $\mathbb{C}[x]/(x^2)$ is finitely generated $\mathbb{C}$-algebra which is not finitely generated $\mathbb{Q}$-algebra since $\mathbb{C}$ is not algebraic over $\mathbb{Q}.$

I want to know is this true in general ? And also want to know what happens if $\mathbb{C}$ is replaced by $\mathbb{R}.$ Thanks.

Answer 1

A non-zero $\mathbb C$-algebra is not denumerable and thus never a finitely generated $\mathbb Q$-algebra.