p-adic integers #

This file defines the p-adic integers ℤ_[p] as the subtype of ℚ_[p] with norm ≤ 1. We show that ℤ_[p]

is complete,
is nonarchimedean,
is a normed ring,
is a local ring, and
is a discrete valuation ring.

The relation between ℤ_[p] and zmod p is established in another file.

Important definitions #

padic_int : the type of p-adic integers

Notation #

We introduce the notation ℤ_[p] for the p-adic integers.

Implementation notes #

Much, but not all, of this file assumes that p is prime. This assumption is inferred automatically by taking [fact p.prime] as a type class argument.

Coercions into ℤ_[p] are set up to work with the norm_cast tactic.

References #

Tags #

p-adic, p adic, padic, p-adic integer

source

def padic_int (p : ℕ) [fact (nat.prime p)] :

Type

The p-adic integers ℤ_[p] are the p-adic numbers with norm ≤ 1.

Equations

ℤ_[p] = {x // ∥x∥ ≤ 1}

Instances for padic_int

Ring structure and coercion to `ℚ_[p]` #

source

@[protected, instance]

def padic_int.padic.has_coe {p : ℕ} [fact (nat.prime p)] :

has_coe ℤ_[p] ℚ_[p]

Equations

padic_int.padic.has_coe = {coe := subtype.val (λ (x : ℚ_[p]), ∥x∥ ≤ 1)}

source

theorem padic_int.ext {p : ℕ} [fact (nat.prime p)] {x y : ℤ_[p]} :

↑x = ↑y → x = y

source

def padic_int.subring (p : ℕ) [fact (nat.prime p)] :

subring ℚ_[p]

The p-adic integers as a subring of ℚ_[p].

Equations

padic_int.subring p = {carrier := {x : ℚ_[p] | ∥x∥ ≤ 1}, mul_mem' := _, one_mem' := _, add_mem' := _, zero_mem' := _, neg_mem' := _}

source

@[simp]

theorem padic_int.mem_subring_iff (p : ℕ) [fact (nat.prime p)] {x : ℚ_[p]} :

x ∈ padic_int.subring p ↔ ∥x∥ ≤ 1

source

@[protected, instance]

def padic_int.has_add {p : ℕ} [fact (nat.prime p)] :

has_add ℤ_[p]

Addition on ℤ_[p] is inherited from ℚ_[p].

Equations

padic_int.has_add = add_mem_class.has_add (padic_int.subring p)

source

@[protected, instance]

def padic_int.has_mul {p : ℕ} [fact (nat.prime p)] :

has_mul ℤ_[p]

Multiplication on ℤ_[p] is inherited from ℚ_[p].

Equations

padic_int.has_mul = mul_mem_class.has_mul (padic_int.subring p)

source

@[protected, instance]

def padic_int.has_neg {p : ℕ} [fact (nat.prime p)] :

has_neg ℤ_[p]

Negation on ℤ_[p] is inherited from ℚ_[p].

Equations

padic_int.has_neg = add_subgroup_class.has_neg

source

@[protected, instance]

def padic_int.has_sub {p : ℕ} [fact (nat.prime p)] :

has_sub ℤ_[p]

Subtraction on ℤ_[p] is inherited from ℚ_[p].

Equations

padic_int.has_sub = add_subgroup_class.has_sub

source

@[protected, instance]

def padic_int.has_zero {p : ℕ} [fact (nat.prime p)] :

has_zero ℤ_[p]

Zero on ℤ_[p] is inherited from ℚ_[p].

Equations

padic_int.has_zero = zero_mem_class.has_zero (padic_int.subring p)

source

@[protected, instance]

def padic_int.inhabited {p : ℕ} [fact (nat.prime p)] :

inhabited ℤ_[p]

Equations

padic_int.inhabited = {default := 0}

source

@[protected, instance]

def padic_int.has_one {p : ℕ} [fact (nat.prime p)] :

has_one ℤ_[p]

One on ℤ_[p] is inherited from ℚ_[p].

Equations

padic_int.has_one = {one := ⟨1, _⟩}

source

@[simp]

theorem padic_int.mk_zero {p : ℕ} [fact (nat.prime p)] {h : ∥0∥ ≤ 1} :

⟨0, h⟩ = 0

source

@[simp]

theorem padic_int.val_eq_coe {p : ℕ} [fact (nat.prime p)] (z : ℤ_[p]) :

z.val = ↑z

source

@[simp, norm_cast]

theorem padic_int.coe_add {p : ℕ} [fact (nat.prime p)] (z1 z2 : ℤ_[p]) :

↑(z1 + z2) = ↑z1 + ↑z2

source

@[simp, norm_cast]

theorem padic_int.coe_mul {p : ℕ} [fact (nat.prime p)] (z1 z2 : ℤ_[p]) :

↑(z1 * z2) = ↑z1 * ↑z2

source

@[simp, norm_cast]

theorem padic_int.coe_neg {p : ℕ} [fact (nat.prime p)] (z1 : ℤ_[p]) :

↑-z1 = -↑z1

source

@[simp, norm_cast]

theorem padic_int.coe_sub {p : ℕ} [fact (nat.prime p)] (z1 z2 : ℤ_[p]) :

↑(z1 - z2) = ↑z1 - ↑z2

source

@[simp, norm_cast]

theorem padic_int.coe_one {p : ℕ} [fact (nat.prime p)] :

↑1 = 1

source

@[simp, norm_cast]

theorem padic_int.coe_zero {p : ℕ} [fact (nat.prime p)] :

↑0 = 0

source

@[protected, instance]

def padic_int.add_comm_group {p : ℕ} [fact (nat.prime p)] :

add_comm_group ℤ_[p]

Equations

padic_int.add_comm_group = non_unital_non_assoc_ring.to_add_comm_group ↥(padic_int.subring p)

source

@[protected, instance]

def padic_int.comm_ring {p : ℕ} [fact (nat.prime p)] :

comm_ring ℤ_[p]

Equations

padic_int.comm_ring = (padic_int.subring p).to_comm_ring

source

@[simp, norm_cast]

theorem padic_int.coe_nat_cast {p : ℕ} [fact (nat.prime p)] (n : ℕ) :

↑↑n = ↑n

source

@[simp, norm_cast]

theorem padic_int.coe_int_cast {p : ℕ} [fact (nat.prime p)] (z : ℤ) :

↑↑z = ↑z

source

def padic_int.coe.ring_hom {p : ℕ} [fact (nat.prime p)] :

ℤ_[p] →+* ℚ_[p]

The coercion from ℤ_[p] to ℚ_[p] as a ring homomorphism.

Equations

padic_int.coe.ring_hom = (padic_int.subring p).subtype

source

@[simp, norm_cast]

theorem padic_int.coe_pow {p : ℕ} [fact (nat.prime p)] (x : ℤ_[p]) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

source

@[simp]

theorem padic_int.mk_coe {p : ℕ} [fact (nat.prime p)] (k : ℤ_[p]) :

⟨↑k, _⟩ = k

source

noncomputable def padic_int.inv {p : ℕ} [fact (nat.prime p)] :

ℤ_[p] → ℤ_[p]

The inverse of a p-adic integer with norm equal to 1 is also a p-adic integer. Otherwise, the inverse is defined to be 0.

Equations

padic_int.inv ⟨k, _x⟩ = dite (∥k∥ = 1) (λ (h : ∥k∥ = 1), ⟨k⁻¹, _⟩) (λ (h : ¬∥k∥ = 1), 0)

source

@[protected, instance]

def padic_int.char_zero {p : ℕ} [fact (nat.prime p)] :

char_zero ℤ_[p]

source

@[simp, norm_cast]

theorem padic_int.coe_int_eq {p : ℕ} [fact (nat.prime p)] (z1 z2 : ℤ) :

↑z1 = ↑z2 ↔ z1 = z2

source

def padic_int.of_int_seq {p : ℕ} [fact (nat.prime p)] (seq : ℕ → ℤ) (h : is_cau_seq (padic_norm p) (λ (n : ℕ), ↑(seq n))) :

ℤ_[p]

A sequence of integers that is Cauchy with respect to the p-adic norm converges to a p-adic integer.

Equations

padic_int.of_int_seq seq h = ⟨⟦⟨λ (n : ℕ), ↑(seq n), h⟩⟧, _⟩

Instances #

We now show that ℤ_[p] is a

complete metric space
normed ring
integral domain

source

@[protected, instance]

noncomputable def padic_int.metric_space (p : ℕ) [fact (nat.prime p)] :

metric_space ℤ_[p]

Equations

padic_int.metric_space p = subtype.metric_space

source

@[protected, instance]

def padic_int.complete_space (p : ℕ) [fact (nat.prime p)] :

complete_space ℤ_[p]

source

@[protected, instance]

noncomputable def padic_int.has_norm (p : ℕ) [fact (nat.prime p)] :

has_norm ℤ_[p]

Equations

padic_int.has_norm p = {norm := λ (z : ℤ_[p]), ∥↑z∥}

source

theorem padic_int.norm_def {p : ℕ} [fact (nat.prime p)] {z : ℤ_[p]} :

∥z∥ = ∥↑z∥

source

@[protected, instance]

noncomputable def padic_int.normed_comm_ring (p : ℕ) [fact (nat.prime p)] :

normed_comm_ring ℤ_[p]

Equations

padic_int.normed_comm_ring p = {to_normed_ring := {to_has_norm := {norm := has_norm.norm (padic_int.has_norm p)}, to_ring := {add := comm_ring.add padic_int.comm_ring, add_assoc := _, zero := comm_ring.zero padic_int.comm_ring, zero_add := _, add_zero := _, nsmul := comm_ring.nsmul padic_int.comm_ring, nsmul_zero' := _, nsmul_succ' := _, neg := comm_ring.neg padic_int.comm_ring, sub := comm_ring.sub padic_int.comm_ring, sub_eq_add_neg := _, zsmul := comm_ring.zsmul padic_int.comm_ring, zsmul_zero' := _, zsmul_succ' := _, zsmul_neg' := _, add_left_neg := _, add_comm := _, int_cast := comm_ring.int_cast padic_int.comm_ring, nat_cast := comm_ring.nat_cast padic_int.comm_ring, one := comm_ring.one padic_int.comm_ring, nat_cast_zero := _, nat_cast_succ := _, int_cast_of_nat := _, int_cast_neg_succ_of_nat := _, mul := comm_ring.mul padic_int.comm_ring, mul_assoc := _, one_mul := _, mul_one := _, npow := comm_ring.npow padic_int.comm_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _}, to_metric_space := {to_pseudo_metric_space := metric_space.to_pseudo_metric_space (padic_int.metric_space p), eq_of_dist_eq_zero := _}, dist_eq := _, norm_mul := _}, mul_comm := _}

source

@[protected, instance]

def padic_int.norm_one_class (p : ℕ) [fact (nat.prime p)] :

norm_one_class ℤ_[p]

source

@[protected, instance]

def padic_int.is_absolute_value (p : ℕ) [fact (nat.prime p)] :

is_absolute_value (λ (z : ℤ_[p]), ∥z∥)

source

@[protected, instance]

def padic_int.is_domain {p : ℕ} [fact (nat.prime p)] :

is_domain ℤ_[p]

Norm #

source

theorem padic_int.norm_le_one {p : ℕ} [fact (nat.prime p)] (z : ℤ_[p]) :

∥z∥ ≤ 1

source

@[simp]

theorem padic_int.norm_mul {p : ℕ} [fact (nat.prime p)] (z1 z2 : ℤ_[p]) :

∥z1 * z2∥ = ∥z1∥ * ∥z2∥

source

@[simp]

theorem padic_int.norm_pow {p : ℕ} [fact (nat.prime p)] (z : ℤ_[p]) (n : ℕ) :

∥z ^ n∥ = ∥z∥ ^ n

source

theorem padic_int.nonarchimedean {p : ℕ} [fact (nat.prime p)] (q r : ℤ_[p]) :

∥q + r∥ ≤ linear_order.max ∥q∥ ∥r∥

source

theorem padic_int.norm_add_eq_max_of_ne {p : ℕ} [fact (nat.prime p)] {q r : ℤ_[p]} :

∥q∥ ≠ ∥r∥ → ∥q + r∥ = linear_order.max ∥q∥ ∥r∥

source

theorem padic_int.norm_eq_of_norm_add_lt_right {p : ℕ} [fact (nat.prime p)] {z1 z2 : ℤ_[p]} (h : ∥z1 + z2∥ < ∥z2∥) :

∥z1∥ = ∥z2∥

source

theorem padic_int.norm_eq_of_norm_add_lt_left {p : ℕ} [fact (nat.prime p)] {z1 z2 : ℤ_[p]} (h : ∥z1 + z2∥ < ∥z1∥) :

∥z1∥ = ∥z2∥

source

@[simp]

theorem padic_int.padic_norm_e_of_padic_int {p : ℕ} [fact (nat.prime p)] (z : ℤ_[p]) :

∥↑z∥ = ∥z∥

source

theorem padic_int.norm_int_cast_eq_padic_norm {p : ℕ} [fact (nat.prime p)] (z : ℤ) :

∥↑z∥ = ∥↑z∥

source

@[simp]

theorem padic_int.norm_eq_padic_norm {p : ℕ} [fact (nat.prime p)] {q : ℚ_[p]} (hq : ∥q∥ ≤ 1) :

∥⟨q, hq⟩∥ = ∥q∥

source

@[simp]

theorem padic_int.norm_p {p : ℕ} [fact (nat.prime p)] :

∥↑p∥ = (↑p)⁻¹

source

@[simp]

theorem padic_int.norm_p_pow {p : ℕ} [fact (nat.prime p)] (n : ℕ) :

∥↑p ^ n∥ = ↑p ^ -↑n

source

@[protected, instance]

def padic_int.complete (p : ℕ) [fact (nat.prime p)] :

cau_seq.is_complete ℤ_[p] has_norm.norm

source

theorem padic_int.exists_pow_neg_lt (p : ℕ) [hp : fact (nat.prime p)] {ε : ℝ} (hε : 0 < ε) :

∃ (k : ℕ), ↑p ^ -↑k < ε

source

theorem padic_int.exists_pow_neg_lt_rat (p : ℕ) [hp : fact (nat.prime p)] {ε : ℚ} (hε : 0 < ε) :

∃ (k : ℕ), ↑p ^ -↑k < ε

source

theorem padic_int.norm_int_lt_one_iff_dvd {p : ℕ} [hp : fact (nat.prime p)] (k : ℤ) :

∥↑k∥ < 1 ↔ ↑p ∣ k

source

theorem padic_int.norm_int_le_pow_iff_dvd {p : ℕ} [hp : fact (nat.prime p)] {k : ℤ} {n : ℕ} :

∥↑k∥ ≤ ↑p ^ -↑n ↔ ↑p ^ n ∣ k

Valuation on `ℤ_[p]` #

source

noncomputable def padic_int.valuation {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) :

ℤ

padic_int.valuation lifts the p-adic valuation on ℚ to ℤ_[p].

Equations

x.valuation = ↑x.valuation

source

theorem padic_int.norm_eq_pow_val {p : ℕ} [hp : fact (nat.prime p)] {x : ℤ_[p]} (hx : x ≠ 0) :

∥x∥ = ↑p ^ -x.valuation

source

@[simp]

theorem padic_int.valuation_zero {p : ℕ} [hp : fact (nat.prime p)] :

0.valuation = 0

source

@[simp]

theorem padic_int.valuation_one {p : ℕ} [hp : fact (nat.prime p)] :

1.valuation = 0

source

@[simp]

theorem padic_int.valuation_p {p : ℕ} [hp : fact (nat.prime p)] :

↑p.valuation = 1

source

theorem padic_int.valuation_nonneg {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) :

0 ≤ x.valuation

source

@[simp]

theorem padic_int.valuation_p_pow_mul {p : ℕ} [hp : fact (nat.prime p)] (n : ℕ) (c : ℤ_[p]) (hc : c ≠ 0) :

(↑p ^ n * c).valuation = ↑n + c.valuation

Units of `ℤ_[p]` #

source

theorem padic_int.mul_inv {p : ℕ} [hp : fact (nat.prime p)] {z : ℤ_[p]} :

∥z∥ = 1 → z * z.inv = 1

source

theorem padic_int.inv_mul {p : ℕ} [hp : fact (nat.prime p)] {z : ℤ_[p]} (hz : ∥z∥ = 1) :

z.inv * z = 1

source

theorem padic_int.is_unit_iff {p : ℕ} [hp : fact (nat.prime p)] {z : ℤ_[p]} :

is_unit z ↔ ∥z∥ = 1

source

theorem padic_int.norm_lt_one_add {p : ℕ} [hp : fact (nat.prime p)] {z1 z2 : ℤ_[p]} (hz1 : ∥z1∥ < 1) (hz2 : ∥z2∥ < 1) :

∥z1 + z2∥ < 1

source

theorem padic_int.norm_lt_one_mul {p : ℕ} [hp : fact (nat.prime p)] {z1 z2 : ℤ_[p]} (hz2 : ∥z2∥ < 1) :

∥z1 * z2∥ < 1

source

@[simp]

theorem padic_int.mem_nonunits {p : ℕ} [hp : fact (nat.prime p)] {z : ℤ_[p]} :

z ∈ nonunits ℤ_[p] ↔ ∥z∥ < 1

source

noncomputable def padic_int.mk_units {p : ℕ} [hp : fact (nat.prime p)] {u : ℚ_[p]} (h : ∥u∥ = 1) :

ℤ_[p]ˣ

A p-adic number u with ∥u∥ = 1 is a unit of ℤ_[p].

Equations

padic_int.mk_units h = let z : ℤ_[p] := ⟨u, _⟩ in {val := z, inv := z.inv, val_inv := _, inv_val := _}

source

@[simp]

theorem padic_int.mk_units_eq {p : ℕ} [hp : fact (nat.prime p)] {u : ℚ_[p]} (h : ∥u∥ = 1) :

↑↑(padic_int.mk_units h) = u

source

@[simp]

theorem padic_int.norm_units {p : ℕ} [hp : fact (nat.prime p)] (u : ℤ_[p]ˣ) :

∥↑u∥ = 1

source

noncomputable def padic_int.unit_coeff {p : ℕ} [hp : fact (nat.prime p)] {x : ℤ_[p]} (hx : x ≠ 0) :

ℤ_[p]ˣ

unit_coeff hx is the unit u in the unique representation x = u * p ^ n. See unit_coeff_spec.

Equations

padic_int.unit_coeff hx = let u : ℚ_[p] := ↑x * ↑p ^ -x.valuation in have hu : ∥u∥ = 1, from _, padic_int.mk_units hu

source

@[simp]

theorem padic_int.unit_coeff_coe {p : ℕ} [hp : fact (nat.prime p)] {x : ℤ_[p]} (hx : x ≠ 0) :

↑(padic_int.unit_coeff hx) = ↑x * ↑p ^ -x.valuation

source

theorem padic_int.unit_coeff_spec {p : ℕ} [hp : fact (nat.prime p)] {x : ℤ_[p]} (hx : x ≠ 0) :

x = ↑(padic_int.unit_coeff hx) * ↑p ^ x.valuation.nat_abs

Various characterizations of open unit balls #

source

theorem padic_int.norm_le_pow_iff_le_valuation {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) (hx : x ≠ 0) (n : ℕ) :

∥x∥ ≤ ↑p ^ -↑n ↔ ↑n ≤ x.valuation

source

theorem padic_int.mem_span_pow_iff_le_valuation {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) (hx : x ≠ 0) (n : ℕ) :

x ∈ ideal.span {↑p ^ n} ↔ ↑n ≤ x.valuation

source

theorem padic_int.norm_le_pow_iff_mem_span_pow {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) (n : ℕ) :

∥x∥ ≤ ↑p ^ -↑n ↔ x ∈ ideal.span {↑p ^ n}

source

theorem padic_int.norm_le_pow_iff_norm_lt_pow_add_one {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) (n : ℤ) :

∥x∥ ≤ ↑p ^ n ↔ ∥x∥ < ↑p ^ (n + 1)

source

theorem padic_int.norm_lt_pow_iff_norm_le_pow_sub_one {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) (n : ℤ) :

∥x∥ < ↑p ^ n ↔ ∥x∥ ≤ ↑p ^ (n - 1)

source

theorem padic_int.norm_lt_one_iff_dvd {p : ℕ} [hp : fact (nat.prime p)] (x : ℤ_[p]) :

∥x∥ < 1 ↔ ↑p ∣ x

source

@[simp]

theorem padic_int.pow_p_dvd_int_iff {p : ℕ} [hp : fact (nat.prime p)] (n : ℕ) (a : ℤ) :

↑p ^ n ∣ ↑a ↔ ↑p ^ n ∣ a

Discrete valuation ring #

source

@[protected, instance]

def padic_int.local_ring {p : ℕ} [hp : fact (nat.prime p)] :

local_ring ℤ_[p]

source

theorem padic_int.p_nonnunit {p : ℕ} [hp : fact (nat.prime p)] :

↑p ∈ nonunits ℤ_[p]

source

theorem padic_int.maximal_ideal_eq_span_p {p : ℕ} [hp : fact (nat.prime p)] :

local_ring.maximal_ideal ℤ_[p] = ideal.span {↑p}

source

theorem padic_int.prime_p {p : ℕ} [hp : fact (nat.prime p)] :

prime ↑p

source

theorem padic_int.irreducible_p {p : ℕ} [hp : fact (nat.prime p)] :

irreducible ↑p

source

@[protected, instance]

def padic_int.discrete_valuation_ring {p : ℕ} [hp : fact (nat.prime p)] :

discrete_valuation_ring ℤ_[p]

source

theorem padic_int.ideal_eq_span_pow_p {p : ℕ} [hp : fact (nat.prime p)] {s : ideal ℤ_[p]} (hs : s ≠ ⊥) :

∃ (n : ℕ), s = ideal.span {↑p ^ n}

source

@[protected, instance]

def padic_int.is_adic_complete {p : ℕ} [hp : fact (nat.prime p)] :

is_adic_complete (local_ring.maximal_ideal ℤ_[p]) ℤ_[p]

p-adic integers #

Important definitions #

Notation #

Implementation notes #

References #

Tags #

Ring structure and coercion to ℚ_[p] #

Instances #

Norm #

Valuation on ℤ_[p] #

Units of ℤ_[p] #

Various characterizations of open unit balls #

Discrete valuation ring #

Ring structure and coercion to `ℚ_[p]` #

Valuation on `ℤ_[p]` #

Units of `ℤ_[p]` #