Proper ideals of quadratic orders are invertible

In an order O of a quadratic number field K, a fractional ideal is proper (its multiplier ring is exactly O) if and only if it is invertible. In higher degree only the implication from invertible to proper survives; the cubic order ℤ + 2ℤ∛2 + 2ℤ∛4 has a proper noninvertible ideal. So for quadratic orders the Picard group Pic O sees every proper ideal class, which is what lets the ring class group of a quadratic order be described by proper ideals or, equivalently, by primitive binary quadratic forms.

The proof is Cox's. Write a nonzero fractional ideal as a lattice I = ℤα + ℤβ and put τ = β / α, a root of a primitive integral quadratic a τ² - b τ - e = 0 (gcd (a, b, e) = 1). Then a τ preserves I, so it lies in O when I is proper. With τ' = b / a - τ the conjugate of τ, the elements a / α and a τ' / α carry I into O, and the products of α, β ∈ I with them are a, b and -e up to integer combinations. Since a, b and e generate the unit ideal of ℤ, the product of I with 1 / I = (O : I) contains 1, so it is the unit ideal.

Main results

In the namespace TauCeti.GlobalNumberFields.NumberFieldOrder:

• IsProperFractionalIdeal.isUnit_of_finrank_eq_two: a proper fractional ideal of an order in a quadratic field is invertible.
• isProperFractionalIdeal_iff_isUnit_of_finrank_eq_two: for an order in a quadratic field, proper and invertible fractional ideals coincide.

References

• D. A. Cox, Primes of the Form x² + ny², §7, Lemma 7.5.
• G. S. Kopp and J. C. Lagarias, Class Field Theory for Orders of Number Fields, §2.

theorem TauCeti.GlobalNumberFields.NumberFieldOrder.IsProperFractionalIdeal.isUnit_of_finrank_eq_two {K : Type u_1} [Field K] [NumberField K] {O : NumberFieldOrder K} (hK : Module.finrank ℚ K = 2) {I : FractionalIdeal (nonZeroDivisors ↥O.toSubalgebra) K} (hI : O.IsProperFractionalIdeal I) : IsUnit I

A proper fractional ideal of a quadratic order is invertible (Cox, Lemma 7.5).

theorem TauCeti.GlobalNumberFields.NumberFieldOrder.isProperFractionalIdeal_iff_isUnit_of_finrank_eq_two {K : Type u_1} [Field K] [NumberField K] (O : NumberFieldOrder K) (hK : Module.finrank ℚ K = 2) (I : FractionalIdeal (nonZeroDivisors ↥O.toSubalgebra) K) : O.IsProperFractionalIdeal I ↔ IsUnit I

Proper and invertible fractional ideals coincide in a quadratic order. In higher degree only invertible ideals are guaranteed to be proper.