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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.

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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.

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