Commit 2023-09-27 14:09 2b8f139e

feat(NumberTheory.NumberField.Units): proof of Dirichlet's unit theorem (#5960) We prove Dirichlet's unit theorem. More precisely, the main results are:

def basisModTorsion : Basis (Fin (Units.rank K)) ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K)))

where Units.rank := Fintype.card (InfinitePlace K) - 1 and

theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! (ζ : torsion K) (e : Fin (Units.rank K) → ℤ),
    x = ζ * ∏ i, (fundSystem K i) ^ (e i)

where fundSystem : Fin (rank K) → (𝓞 K)ˣ is a fundamental system of units. The exponents in exist_unique_eq_mul_prod can be computed via the following result:

theorem fun_eq_repr {x ζ : (𝓞 K)ˣ} {f : Fin (rank K) → ℤ} (hζ : ζ ∈ torsion K) 
    (h : x = ζ * ∏ i, (fundSystem K i) ^ (f i)) : f = (basisModTorsion K).repr (Additive.ofMul ↑x)