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| 1 | +/- |
| 2 | +Copyright (c) 2021 Anne Baanen. All rights reserved. |
| 3 | +Released under Apache 2.0 license as described in the file LICENSE. |
| 4 | +Authors: Anne Baanen |
| 5 | +-/ |
| 6 | +import number_theory.class_number.admissible_card_pow_degree |
| 7 | +import number_theory.class_number.finite |
| 8 | +import number_theory.function_field |
| 9 | + |
| 10 | +/-! |
| 11 | +# Class numbers of function fields |
| 12 | +
|
| 13 | +This file defines the class number of a function field as the (finite) cardinality of |
| 14 | +the class group of its ring of integers. It also proves some elementary results |
| 15 | +on the class number. |
| 16 | +
|
| 17 | +## Main definitions |
| 18 | +- `function_field.class_number`: the class number of a function field is the (finite) |
| 19 | +cardinality of the class group of its ring of integers |
| 20 | +-/ |
| 21 | + |
| 22 | +namespace function_field |
| 23 | + |
| 24 | +variables (Fq F : Type) [field Fq] [fintype Fq] [field F] |
| 25 | +variables [algebra (polynomial Fq) F] [algebra (ratfunc Fq) F] |
| 26 | +variables [is_scalar_tower (polynomial Fq) (ratfunc Fq) F] |
| 27 | +variables [function_field Fq F] [is_separable (ratfunc Fq) F] |
| 28 | + |
| 29 | +open_locale classical |
| 30 | + |
| 31 | +namespace ring_of_integers |
| 32 | + |
| 33 | +open function_field |
| 34 | + |
| 35 | +noncomputable instance : fintype (class_group (ring_of_integers Fq F) F) := |
| 36 | +class_group.fintype_of_admissible_of_finite (ratfunc Fq) F |
| 37 | + (polynomial.card_pow_degree_is_admissible : absolute_value.is_admissible |
| 38 | + (polynomial.card_pow_degree : absolute_value (polynomial Fq) ℤ)) |
| 39 | + |
| 40 | +end ring_of_integers |
| 41 | + |
| 42 | +/-- The class number in a function field is the (finite) cardinality of the class group. -/ |
| 43 | +noncomputable def class_number : ℕ := fintype.card (class_group (ring_of_integers Fq F) F) |
| 44 | + |
| 45 | +/-- The class number of a function field is `1` iff the ring of integers is a PID. -/ |
| 46 | +theorem class_number_eq_one_iff : |
| 47 | + class_number Fq F = 1 ↔ is_principal_ideal_ring (ring_of_integers Fq F) := |
| 48 | +card_class_group_eq_one_iff |
| 49 | + |
| 50 | +end function_field |
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