Rep herbrand Quotient is Nonarchimedean Local Field units
Rep.herbrandQuotient_isNonarchimedeanLocalField_units
Plain-language statement
herbrand quotient of Lˣ is [L:K]
Exact Lean statement
theorem Rep.herbrandQuotient_isNonarchimedeanLocalField_units :
herbrandQuotient
((Rep.resFunctor (IsGaloisGroup.mulEquivAlgEquiv G K L).toMonoidHom).obj
(Rep.ofAlgebraAutOnUnits K L) : Rep ℤ G)
= Module.finrank K LFormal artifact
Lean source
theorem Rep.herbrandQuotient_isNonarchimedeanLocalField_units : herbrandQuotient ((Rep.resFunctor (IsGaloisGroup.mulEquivAlgEquiv G K L).toMonoidHom).obj (Rep.ofAlgebraAutOnUnits K L) : Rep ℤ G) = Module.finrank K L := by have := Fintype.ofFinite G have h1 : (valuationShortComplex G K L).X₁.herbrandQuotient = 1 := Rep.herbrandQuotient_isNonarchimedeanLocalField_integer_units K L G have h1' : (valuationShortComplex G K L).X₁.herbrandQuotient ≠ 0 := by simp [h1] have h3 : (valuationShortComplex G K L).X₃.herbrandQuotient = Nat.card G := Rep.herbrandQuotient_trivial_int_eq_card G have h3' : (valuationShortComplex G K L).X₃.herbrandQuotient ≠ 0 := by simp [h3] have := herbrandQuotient_eq_of_shortExact valuationShortComplex.shortExact h1' ?_ h3' · convert! this simp [h1, h3, -Nat.card_eq_fintype_card, IsGaloisGroup.card_eq_finrank G K L] · apply Rep.herbrandQuotient_ne_zero_of_shortExact₂ valuationShortComplex.shortExact h1' h3'- Project
- Class Field Theory
- License
- Apache-2.0
- Commit
- f18cd7fd1575
- Source
- ClassFieldTheory/IsNonarchimedeanLocalField/HerbrandQuotient.lean:35-53
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