This follows from a combination of Lemmas \ref{lemma-lci-global} and \ref{lemma-lci-field-change-local}. But we also give a different proof here (based on the same principles). \medskip\noindent Set $S' = S \otimes_k K$. Let $\alpha : k[x_1, \ldots, x_n] \to S$ be a presentation with kernel $I$. Let $\alpha' : K[x_1, \ldots, x_n] \to S'$ be the induced presentation with kernel $I'$. \medskip\noindent Suppose that $S$ is a local complete intersection. Pick a prime $\mathfrak q \subset S'$. Denote $\mathfrak q'$ the corresponding prime of $K[x_1, \ldots, x_n]$, $\mathfrak p$ the corresponding prime of $S$, and $\mathfrak p'$ the corresponding prime of $k[x_1, \ldots, x_n]$. Consider the following diagram of Noetherian local rings $$ \xymatrix{ S'_{\mathfrak q} &  K[x_1, \ldots, x_n]_{\mathfrak q'} \ar[l] \\ S_{\mathfrak p}\ar[u] &  k[x_1, \ldots, x_n]_{\mathfrak p'} \ar[u] \ar[l] } $$ By Lemma \ref{lemma-lci} we know that $S_{\mathfrak p}$ is cut out by some regular sequence $f_1, \ldots, f_c$ in $k[x_1, \ldots, x_n]_{\mathfrak p'}$. Since the right vertical arrow is flat we see that the images of $f_1, \ldots, f_c$ form a regular sequence in $K[x_1, \ldots, x_n]_{\mathfrak q'}$. Because tensoring with $K$ over $k$ is an exact functor we have $S'_{\mathfrak q} = K[x_1, \ldots, x_n]_{\mathfrak q'}/(f_1, \ldots, f_c)$. Hence by Lemma \ref{lemma-lci} again we see that $S'$ is a local complete intersection in a neighbourhood of $\mathfrak q$. Since $\mathfrak q$ was arbitrary we see that $S'$ is a local complete intersection over $K$. \medskip\noindent Suppose that $S'$ is a local complete intersection. Pick a maximal ideal $\mathfrak m$ of $S$. Let $\mathfrak m'$ denote the corresponding maximal ideal of $k[x_1, \ldots, x_n]$. Denote $\kappa = \kappa(\mathfrak m)$ the residue field. By Remark \ref{remark-fundamental-diagram} the primes of $S'$ lying over $\mathfrak m$ correspond to primes in $K \otimes_k \kappa$. By the Hilbert-Nullstellensatz Theorem \ref{theorem-nullstellensatz} we have $[\kappa : k] < \infty$. Hence $K \otimes_k \kappa$ is finite nonzero over $K$. Hence $K \otimes_k \kappa$ has a finite number $> 0$ of primes which are all maximal, each of which has a residue field finite over $K$ (see Section \ref{section-artinian}). Hence there are finitely many $> 0$ prime ideals $\mathfrak n \subset S'$ lying over $\mathfrak m$, each of which is maximal and  has a residue field which is finite over $K$. Pick one, say $\mathfrak n \subset S'$, and let $\mathfrak n' \subset K[x_1, \ldots, x_n]$ denote the corresponding prime ideal of $K[x_1, \ldots, x_n]$. Note that since $V(\mathfrak mS')$ is finite, we see that $\mathfrak n$ is an isolated closed point of it, and we deduce that $\mathfrak mS'_{\mathfrak n}$ is an ideal of definition of $S'_{\mathfrak n}$. This implies that $\dim(S_{\mathfrak m}) = \dim(S'_{\mathfrak n})$ for example by Lemma \ref{lemma-dimension-base-fibre-equals-total}. (This can also be seen using Lemma \ref{lemma-dimension-at-a-point-preserved-field-extension}.) Consider the corresponding diagram of Noetherian local rings $$ \xymatrix{ S'_{\mathfrak n} &  K[x_1, \ldots, x_n]_{\mathfrak n'} \ar[l] \\ S_{\mathfrak m}\ar[u] &  k[x_1, \ldots, x_n]_{\mathfrak m'} \ar[u] \ar[l] } $$ According to Lemma \ref{lemma-change-base-NL} we have $\NL(\alpha) \otimes_S S' = \NL(\alpha')$, in particular $I'/(I')^2 = I/I^2 \otimes_S S'$. Thus $(I/I^2)_{\mathfrak m} \otimes_{S_{\mathfrak m}} \kappa$ and $(I'/(I')^2)_{\mathfrak n} \otimes_{S'_{\mathfrak n}} \kappa(\mathfrak n)$ have the same dimension. Since $(I'/(I')^2)_{\mathfrak n}$ is free of rank $n - \dim S'_{\mathfrak n}$ we deduce that $(I/I^2)_{\mathfrak m}$ can be generated by $n - \dim S'_{\mathfrak n} = n - \dim S_{\mathfrak m}$ elements. By Lemma \ref{lemma-lci} we see that $S$ is a local complete intersection in a neighbourhood of $\mathfrak m$. Since $\mathfrak m$ was any maximal ideal we conclude that $S$ is a local complete intersection.