Inducing one point closed subset with a closed subscheme structure so that the stalk of the subscheme is a field

Let $$(X,\mathcal O_X)$$ be a Noetherian scheme. Let $$x\in X$$ be a closed point and $$Y:=\{x \}$$ .

Is it always possible to make $$Y$$ into a scheme such that $$(Y, \mathcal O_Y)$$ is a closed subscheme of $$(X, \mathcal O_X )$$ and $$\mathcal O_{Y,x}$$ is a field ?

If this is indeed true, then can we have the same result if more generally we started with some $$x \in X$$ and set $$Y=\overline { \{x \} }$$ ?

EDIT: Some thoughts: Let $$x \in X$$ and $$Y:=\overline { \{x \} }$$. Since $$\{x\}$$ is irreducible in $$X$$, hence so is its closure $$Y$$. Let $$\mathcal O_Y$$ be a structure sheaf on $$Y$$ , then since $$x \in Y$$ is generic for $$Y$$ and $$Y$$ is irreducible, so $$\dim \mathcal O_{Y,x}=0$$. So to find a closed subscheme structure $$(Y, \mathcal O_Y)$$ such that $$\mathcal O_{Y,x}$$ is a field, we just need to ensure $$\mathcal O_{Y,x}$$ is an integral domain.

Also note that the answer to both the questions are positive if $$X$$ is affine.

In both cases the answer is yes. Put the reduced induced scheme structure on $$Y$$ - then $$Y$$ is irreducible and reduced, thus integral, so the stalk of $$\mathcal{O}_Y$$ at $$x$$, the generic point of $$Y$$, is a field.