FunctionField.classNumber_eq_one_iff

theorem FunctionField.classNumber_eq_one_iff : ∀ (Fq F : Type) [inst : Field Fq] [inst_1 : Fintype Fq] [inst_2 : Field F] [inst_3 : Algebra (Polynomial Fq) F] [inst_4 : Algebra (RatFunc Fq) F] [inst_5 : IsScalarTower (Polynomial Fq) (RatFunc Fq) F] [inst_6 : FunctionField Fq F] [inst_7 : IsSeparable (RatFunc Fq) F], FunctionField.classNumber Fq F = 1 ↔ IsPrincipalIdealRing ↥(FunctionField.ringOfIntegers Fq F)

The theorem `FunctionField.classNumber_eq_one_iff` states that for any function field `F`, defined over a finite field `Fq`, the class number of the function field is equal to one if and only if the ring of integers of the function field is a principal ideal ring. Here, the class number provides a measure of the "complexity" or "degree of non-commutativity" of the ring of integers in the function field, while a principal ideal ring is a type of ring where all ideals are generated by a single element. This theorem provides a connection between these two seemingly disparate mathematical concepts.

More concisely: The class number of a function field equals 1 if and only if its ring of integers is a principal ideal ring.