A Noetherian complete local ring is J-2 (More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-J-2}), Nagata (Algebra, Proposition \ref{algebra-proposition-ubiquity-nagata}), excellent (More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-excellent}), and has a dualizing complex (Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing}). Moreover, the same is true for any ring essentially of finite type over $A$. If $B$ is a excellent local normal domain, then the completion $B^\wedge$ is normal (as $B \to B^\wedge$ is regular and More on Algebra, Lemma \ref{more-algebra-lemma-normal-goes-up} applies). In other words, the local rings which we encounter in the rest of the proof will have the required ``excellency'' properties required of them. \medskip\noindent Choose $A_0 \subset A$ with $A_0$ a regular complete local ring and $A_0 \to A$ finite, see Algebra, Lemma \ref{algebra-lemma-complete-local-Noetherian-domain-finite-over-regular}. This induces a finite extension of fraction fields $K_0 \subset K$. We will argue by induction on $[K : K_0]$. The base case is when the degree is $1$ in which case $A_0 = A$ and the result is true. \medskip\noindent Suppose there is an intermediate field $K_0 \subset L \subset K$, $K_0 \not = L \not = K$. Let $B \subset A$ be the integral closure of $A_0$ in $L$. By induction we choose a resolution of singularities $Y \to \Spec(B)$. Let $X$ be the normalization of $Y \times_{\Spec(B)} \Spec(A)$. Picture: $$ \xymatrix{ X \ar[r] \ar[d] & \Spec(A) \ar[d] \\ Y \ar[r] & \Spec(B) } $$ Since $A$ is J-2 the regular locus of $X$ is open. Since $X$ is a normal surface we conclude that $X$ has at worst finitely many singular points $x_1, \ldots, x_n$ which are closed points with $\dim(\mathcal{O}_{X, x_i}) = 2$. For each $i$ let $y_i \in Y$ be the image. Since $\mathcal{O}_{Y, y_i}^\wedge \to \mathcal{O}_{X, x_i}^\wedge$ is finite of smaller degree than before we conclude by induction hypothesis that $\mathcal{O}_{X, x_i}^\wedge$ has resolution of singularities. By Lemma \ref{lemma-existence-implies-existence-by-normalized-blowing-ups} there is a sequence $$ Z^\wedge_{i, n_i} \to \ldots \to Z^\wedge_{i, 1} \to \Spec(\mathcal{O}_{X, x_i}^\wedge) $$ of normalized blowups with $Z^\wedge_{i, n_i}$ regular. By Lemma \ref{lemma-normalized-blowup-completion} there is a corresponding sequence of normalized blowing ups $$ Z_{i, n_i} \to \ldots \to Z_{i, 1} \to \Spec(\mathcal{O}_{X, x_i}) $$ Then $Z_{i, n_i}$ is a regular scheme by Lemma \ref{lemma-port-regularity-to-completion}. By Lemma \ref{lemma-equivalence-sequence-normalized-blowups} we can fit these normalized blowing ups into a corresponding sequence $$ Z_n \to Z_{n - 1} \to \ldots \to Z_1 \to X $$ and of course $Z_n$ is regular too (look at the local rings). This proves the induction step. \medskip\noindent Assume there is no intermediate field $K_0 \subset L \subset K$ with $K_0 \not = L \not = K$. Then either $K/K_0$ is separable or the characteristic to $K$ is $p$ and $[K : K_0] = p$. Then either Lemma \ref{lemma-go-up-separable} or \ref{lemma-go-up-degree-p} implies that reduction to rational singularities is possible. By Lemma \ref{lemma-reduce-to-rational} we conclude that there exists a normal modification $X \to \Spec(A)$ such that for every singular point $x$ of $X$ the local ring $\mathcal{O}_{X, x}$ defines a rational singularity. Since $A$ is J-2 we find that $X$ has finitely many singular points $x_1, \ldots, x_n$. By Lemma \ref{lemma-rational-to-gorenstein} there exists a finite sequence of blowups in singular closed points $$ X_{i, n_i} \to X_{i, n_i - 1} \to \ldots \to \Spec(\mathcal{O}_{X, x_i}) $$ such that $X_{i, n_i}$ is Gorenstein, i.e., has an invertible dualizing module. By (the essentially trivial) Lemma \ref{lemma-equivalence-sequence-blowups} with $n = \sum n_a$ these sequences correspond to a sequence of blowups $$ X_n \to X_{n - 1} \to \ldots \to X $$ such that $X_n$ is normal and the local rings of $X_n$ are Gorenstein. Again $X_n$ has a finite number of singular points $x'_1, \ldots, x'_s$, but this time the singularities are rational double points, more precisely, the local rings $\mathcal{O}_{X_n, x'_i}$ are as in Lemma \ref{lemma-resolve-rational-double-points}. Arguing exactly as above we conclude that the lemma is true.