suffices φ (galConj K p ζ) = conj (φ ζ)
  by
  rw [← Function.funext_iff]
  congr
  rw [DFunLike.coe_fn_eq]
  apply (hζ.powerBasis ℚ).rat_hom_ext
  exact this.symm
K : Type u_1
p : ℕ+
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
φ : K →+* ℂ
⊢ ∀ (x : K), ((starRingEnd ℂ).comp φ) x = (φ.comp ↑(galConj K p)) x
K : Type u_1
p : ℕ+
inst✝² : Field K
inst✝¹ : CharZero K
inst✝ : IsCyclotomicExtension {p} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑p
φ : K →+* ℂ
⊢ φ ((galConj K p) ζ) = (starRingEnd ℂ) (φ ζ)