# On Artin-Hironaka lemma and Galois theory

Let $A=k[[t]]$ Let $B$ a flat $A$-finite algebra which is etale and Galois at the generic point.

Then by Artin lemma 3.12 (ii) in his IHES paper on approximation, we know that there exists an integer $c$ such that:

if we have an other $A$-finite flat algebra generically étale and Galois $B'$ such that $B/\pi^{c}\cong B'/\pi^{c}$ implies $B\cong B'$.

Can we be more precise about the value of $c$ for exemple how can we relate it to $B$?

Moreover, can we get a isomorphism $\phi: B\cong B'$ which lifts the isomorphism mod $\pi^{c}$?

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