# mathlibdocumentation

number_theory.function_field

# Function fields #

This file defines a function field and the ring of integers corresponding to it.

## Main definitions #

• function_field Fq F states that F is a function field over the (finite) field Fq, i.e. it is a finite extension of the field of rational functions in one variable over Fq.
• function_field.ring_of_integers defines the ring of integers corresponding to a function field as the integral closure of polynomial Fq in the function field.

## Implementation notes #

The definitions that involve a field of fractions choose a canonical field of fractions, but are independent of that choice. We also omit assumptions like finite Fq or is_scalar_tower (polynomial Fq) (fraction_ring (polynomial Fq)) F in definitions, adding them back in lemmas when they are needed.

## Tags #

function field, ring of integers

def function_field (Fq F : Type) [field Fq] [field F] [algebra (ratfunc Fq) F] :
Prop

F is a function field over the finite field Fq if it is a finite extension of the field of rational functions in one variable over Fq.

Note that F can be a function field over multiple, non-isomorphic, Fq.

@[protected]
theorem function_field_iff (Fq F : Type) [field Fq] [field F] (Fqt : Type u_1) [field Fqt] [algebra (polynomial Fq) Fqt] [ Fqt] [algebra (ratfunc Fq) F] [algebra Fqt F] [algebra (polynomial Fq) F] [ Fqt F] [ (ratfunc Fq) F] :
F F

F is a function field over Fq iff it is a finite extension of Fq(t).

noncomputable def function_field.ring_of_integers (Fq F : Type) [field Fq] [field F] [algebra (polynomial Fq) F] :

The function field analogue of number_field.ring_of_integers: function_field.ring_of_integers Fq Fqt F is the integral closure of Fq[t] in F.

We don't actually assume F is a function field over Fq in the definition, only when proving its properties.

Equations
@[protected, instance]
@[protected, instance]
@[protected, instance]
def function_field.ring_of_integers.is_fraction_ring (Fq F : Type) [field Fq] [field F] [algebra (polynomial Fq) F] [algebra (ratfunc Fq) F] [ F] [ (ratfunc Fq) F] :
@[protected, instance]
def function_field.ring_of_integers.is_integrally_closed (Fq F : Type) [field Fq] [field F] [algebra (polynomial Fq) F] [algebra (ratfunc Fq) F] [ F] [ (ratfunc Fq) F] :
@[protected, instance]
def function_field.ring_of_integers.is_dedekind_domain (Fq F : Type) [field Fq] [field F] [algebra (polynomial Fq) F] [algebra (ratfunc Fq) F] [ F] [ (ratfunc Fq) F] [is_separable (ratfunc Fq) F] :