Canonical resolutions over non-algebraically-closed ground fields

Galois equivariance. The positive marking is unchanged by base change, and generic nonvanishing persists: the field extension $K\subseteq \bar{K}$ is flat, and each generic point of a component of $X_{\bar{K}}$ lies over a generic point of a component of $X$, where $\mathcal{I}$ is the unit ideal. Thus the base-changed marked ideal satisfies the proposition's hypotheses. By [F2] each $\sigma \in G$ maps the canonical resolution $({\bar{X}}_i)$ with its centers ${\bar{C}}_i$ to the canonical resolution of the same marked ideal; by uniqueness of the canonical resolution (its invariants are intrinsic) this conjugate resolution agrees with the original one, so the centers ${\bar{C}}_i$ and the invariant strata are $G$-stable.

Prove descent on an affine $\mathrm{Spec}A\subseteq X_i$ defined over $K$, with ideal $J_{\bar{K}}$ of the invariant center. For $f\in J_{\bar{K}}$ choose a finite-dimensional $K$-subspace $V\subseteq A$ containing its finitely many coefficient vectors, and a finite Galois subextension $L\mathrm{/}K$ containing its scalar coefficients (adjoin the finitely many roots of their minimal polynomials). The space $W=J_{\bar{K}}\cap (V{\otimes }_{K}L)$ is stable under $\mathrm{Gal}(L\mathrm{/}K)$ by [F6]. By [F5], $f$ is an $L$-linear combination of vectors in $W^{\mathrm{Gal}(L\mathrm{/}K)}\subseteq V\subseteq A$. These invariant vectors belong to $J:=J_{\bar{K}}\cap A$, so $J_{\bar{K}}=J(A{\otimes }_K\bar{K})$. The ideal $J$ is finitely generated because $A$ is Noetherian. Intersections with $A$ commute with localization: if a localized class lies in the extended ideal, some power of its denominator multiplies its numerator into that ideal and hence into $J$. Thus these affine ideals glue uniquely. By [F7] their quotient rings define smooth centers $C_i$ whose scalar extensions are the original ${\bar{C}}_i$.

The resolution over $K$. The blowup construction commutes with the faithfully flat base change $K\to \bar{K}$ (Flat base change for blowups, and failure without flatness), so the sequence $X_{i+1}={\mathrm{Bl}}_{C_i}{X}_i$ over $K$ base changes to ${\bar{X}}_{i+1}$; by construction the supports satisfy $\mathrm{supp}({\mathcal{I}}_i,\mu {)}_{\bar{K}}=\mathrm{supp}({\bar{\mathcal{I}}}_i,\mu )$ by the derivative-support equality in characteristic zero, since derivative ideals commute with separable algebraic scalar extension; at the last stage both are empty, hence $\mathrm{supp}({\mathcal{I}}_m,\mu )=\mathrm{\varnothing }$ over $K$ and the sequence is a resolution of $(\mathcal{I},E,\mu )$ over $K$. The closed superlevels of $\mathrm{inv}$ and the lexicographic pairs $(\mathrm{inv},\nu )$ and $(\mathrm{inv},\rho )$ are $G$-stable by step 1.1 and descend by the same ideal argument in step 2.1. Their finite ranges reconstruct unique functions over $K$: finite differences of primary superlevels determine $\{\mathrm{inv}=a\}$, and restricting the descended pair thresholds $(a,b)$ to this stratum gives its auxiliary superlevels. These sets determine each auxiliary value, are relatively closed on that stratum, and pull back to the original level sets. The descended primary and pair superlevels are closed, preserving the center maxima and every pointwise descent comparison. Thus the resolution is canonical, determined by the intrinsic resolution over $\bar{K}$; and it commutes with smooth morphisms and embeddings over $K$ by applying [F4] after base change to $\bar{K}$ and descending the equal center ideals by step 2.1. The blowups and their natural comparison maps are then defined over $K$ by their Rees-algebra constructions. SNC of the boundary and center descends as well: the individual divisors and their intersection quotients are smooth after scalar extension, and the exact codimensions are preserved by field extension, giving the strict normal crossings criterion (Stacks, tag 0BIA).