Exercise 111.55.10. Let $k$ be an algebraically closed field. Let $A \subset B$ be an extension of domains which are both finite type $k$-algebras. Prove that the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ contains a nonempty open subset of $\mathop{\mathrm{Spec}}(A)$ using the following steps:

Prove it if $A \to B$ is also finite.

Prove it in case the fraction field of $B$ is a finite extension of the fraction field of $A$.

Reduce the statement to the previous case.

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