Suppose $X$ is a projective scheme (or you can only consider the hypersurface case) in $\mathbb{P}^n$ over an algebraic closed field $k$, and $K/k$ is a field extension. Let $\xi$ be a point (neither closed point nor generic point) in $X$. Does the multiplicity of $x$ change under the base change from $k$ to $K$? I mean the multiplicities of the original images of $\xi$ in $X\times\_kK$.
Here we use the multiplicity defined by Hilbert-Shamuel funtion.



Whether the multiplicity changes under a fields extension
Take a irreducible component of $\xi\times\_kK$. If $k$ is perfect, the multiplicity of any of the irreducible component of $\xi\times\_kK$ is not change.


