English
Equality in the PerfectClosure Galois group is determined by Frobenius iterations on the respective components.
Русский
Равенство в группе Галуа PerfectClosure определяется итерациями Фробениуса на соответствующих компонентах.
LaTeX
$$$\mathrm{eq\_iff}:\; \forall (K) [Field K] [IsReduced K] (p) [Fact (Nat.Prime p)] [CharP K p]\ (x,y:\mathbb{N}\times K),\ mk(K,p)(x)=mk(K,p)(y)\iff (\mathrm{frobenius}(K,p))^{[y.1]}x.2=(\mathrm{frobenius}(K,p))^{[x.1]}y.2.$$$
Lean4
/-- If `p` splits in `F` then the `p.gal` is trivial. -/
def uniqueGalOfSplits (h : p.Splits (RingHom.id F)) : Unique p.Gal
where
default := 1
uniq
f :=
AlgEquiv.ext fun x =>
by
obtain ⟨y, rfl⟩ :=
Algebra.mem_bot.mp ((SetLike.ext_iff.mp ((IsSplittingField.splits_iff _ p).mp h) x).mp Algebra.mem_top)
rw [AlgEquiv.commutes, AlgEquiv.commutes]
