# mathlibdocumentation

number_theory.number_field

# Number fields #

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

## Main definitions #

• number_field defines a number field as a field which has characteristic zero and is finite dimensional over ℚ.
• ring_of_integers defines the ring of integers (or number ring) corresponding to a number field as the integral closure of ℤ in the number field.

## Implementation notes #

The definitions that involve a field of fractions choose a canonical field of fractions, but are independent of that choice.

## Tags #

number field, ring of integers

@[class]
structure number_field (K : Type u_1) [field K] :
Prop
• to_char_zero :
• to_finite_dimensional :

A number field is a field which has characteristic zero and is finite dimensional over ℚ.

Instances
theorem number_field.is_algebraic (K : Type u_1) [field K] [nf : number_field K] :
def number_field.ring_of_integers (K : Type u_1) [field K] :

The ring of integers (or number ring) corresponding to a number field is the integral closure of ℤ in the number field.

Equations
@[instance]
@[instance]
theorem number_field.ring_of_integers.is_integral_coe {K : Type u_1} [field K] (x : ) :
def number_field.ring_of_integers.equiv {K : Type u_1} [field K] (R : Type u_2) [comm_ring R] [ K] [ K] :

The ring of integers of K are equivalent to any integral closure of ℤ in K

Equations
@[instance]
@[instance]

The ring of integers of ℚ as a number field is just ℤ.

Equations