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ring_theory.discrete_valuation_ring

Discrete valuation rings #

This file defines discrete valuation rings (DVRs) and develops a basic interface for them.

Important definitions #

There are various definitions of a DVR in the literature; we define a DVR to be a local PID which is not a field (the first definition in Wikipedia) and prove that this is equivalent to being a PID with a unique non-zero prime ideal (the definition in Serre's book "Local Fields").

Let R be an integral domain, assumed to be a principal ideal ring and a local ring.

Definitions #

Implementation notes #

It's a theorem that an element of a DVR is a uniformizer if and only if it's irreducible. We do not hence define uniformizer at all, because we can use irreducible instead.

Tags #

discrete valuation ring

@[class]
structure discrete_valuation_ring (R : Type u) [integral_domain R] :
Prop

An integral domain is a discrete valuation ring (DVR) if it's a local PID which is not a field.

Instances

An element of a DVR is irreducible iff it is a uniformizer, that is, generates the maximal ideal of R

Uniformisers exist in a DVR

theorem discrete_valuation_ring.exists_prime (R : Type u) [integral_domain R] [discrete_valuation_ring R] :
∃ (ϖ : R), prime ϖ

Uniformisers exist in a DVR

an integral domain is a DVR iff it's a PID with a unique non-zero prime ideal

Alternative characterisation of discrete valuation rings.

Equations

An integral domain in which there is an irreducible element p such that every nonzero element is associated to a power of p is a unique factorization domain. See discrete_valuation_ring.of_has_unit_mul_pow_irreducible_factorization.

theorem discrete_valuation_ring.of_ufd_of_unique_irreducible {R : Type u} [integral_domain R] [unique_factorization_monoid R] (h₁ : ∃ (p : R), irreducible p) (h₂ : ∀ ⦃p q : R⦄, irreducible pirreducible qassociated p q) :

A unique factorization domain with at least one irreducible element in which all irreducible elements are associated is a discrete valuation ring.

An integral domain in which there is an irreducible element p such that every nonzero element is associated to a power of p is a discrete valuation ring.

theorem discrete_valuation_ring.associated_pow_irreducible {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] {x : R} (hx : x 0) {ϖ : R} (hirr : irreducible ϖ) :
∃ (n : ), associated x ^ n)
theorem discrete_valuation_ring.eq_unit_mul_pow_irreducible {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] {x : R} (hx : x 0) {ϖ : R} (hirr : irreducible ϖ) :
∃ (n : ) (u : units R), x = (u) * ϖ ^ n
theorem discrete_valuation_ring.ideal_eq_span_pow_irreducible {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] {s : ideal R} (hs : s ) {ϖ : R} (hirr : irreducible ϖ) :
∃ (n : ), s = ideal.span ^ n}
theorem discrete_valuation_ring.unit_mul_pow_congr_pow {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] {p q : R} (hp : irreducible p) (hq : irreducible q) (u v : units R) (m n : ) (h : (u) * p ^ m = (v) * q ^ n) :
m = n
theorem discrete_valuation_ring.unit_mul_pow_congr_unit {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] {ϖ : R} (hirr : irreducible ϖ) (u v : units R) (m n : ) (h : (u) * ϖ ^ m = (v) * ϖ ^ n) :
u = v

The additive valuation on a DVR #

theorem discrete_valuation_ring.add_val_def {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] (r : R) (u : units R) {ϖ : R} (hϖ : irreducible ϖ) (n : ) (hr : r = (u) * ϖ ^ n) :
theorem discrete_valuation_ring.add_val_def' {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] (u : units R) {ϖ : R} (hϖ : irreducible ϖ) (n : ) :
theorem irreducible.add_val_pow {R : Type u_1} [integral_domain R] [discrete_valuation_ring R] {ϖ : R} (h : irreducible ϖ) (n : ) :