Documentation

Mathlib.RingTheory.Perfection

Ring Perfection and Tilt #

In this file we define the perfection of a ring of characteristic p, and the tilt of a field given a valuation to ℝ≥0.

TODO #

Define the valuation on the tilt, and define a characteristic predicate for the tilt.

def Monoid.perfection (M : Type u₁) [CommMonoid M] (p : ) :
Submonoid (M)

The perfection of a monoid M, defined to be the projective limit of M using the p-th power maps M → M indexed by the natural numbers, implemented as { f : ℕ → M | ∀ n, f (n + 1) ^ p = f n }.

Equations
def Ring.perfectionSubsemiring (R : Type u₁) [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] :

The perfection of a ring R with characteristic p, as a subsemiring, defined to be the projective limit of R using the Frobenius maps R → R indexed by the natural numbers, implemented as { f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }.

Equations
def Ring.perfectionSubring (R : Type u₁) [CommRing R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] :
Subring (R)

The perfection of a ring R with characteristic p, as a subring, defined to be the projective limit of R using the Frobenius maps R → R indexed by the natural numbers, implemented as { f : ℕ → R | ∀ n, f (n + 1) ^ p = f n }.

Equations
def Ring.Perfection (R : Type u₁) [CommSemiring R] (p : ) :
Type u₁

The perfection of a ring R with characteristic p, defined to be the projective limit of R using the Frobenius maps R → R indexed by the natural numbers, implemented as {f : ℕ → R // ∀ n, f (n + 1) ^ p = f n}.

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instance Perfection.charP (R : Type u₁) [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] :
instance Perfection.ring (p : ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommRing R] [CharP R p] :
Equations
def Perfection.coeff (R : Type u₁) [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] (n : ) :

The n-th coefficient of an element of the perfection.

Equations
theorem Perfection.ext {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] {f g : Ring.Perfection R p} (h : ∀ (n : ), (coeff R p n) f = (coeff R p n) g) :
f = g
theorem Perfection.ext_iff {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] {f g : Ring.Perfection R p} :
f = g ∀ (n : ), (coeff R p n) f = (coeff R p n) g
def Perfection.pthRoot (R : Type u₁) [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] :

The p-th root of an element of the perfection.

Equations
@[simp]
theorem Perfection.coeff_mk {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : R) (hf : ∀ (n : ), f (n + 1) ^ p = f n) (n : ) :
(coeff R p n) f, hf = f n
theorem Perfection.coeff_pthRoot {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ) :
(coeff R p n) ((pthRoot R p) f) = (coeff R p (n + 1)) f
theorem Perfection.coeff_pow_p {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ) :
(coeff R p (n + 1)) (f ^ p) = (coeff R p n) f
theorem Perfection.coeff_pow_p' {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ) :
(coeff R p (n + 1)) f ^ p = (coeff R p n) f
theorem Perfection.coeff_frobenius {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n : ) :
(coeff R p (n + 1)) ((frobenius (Ring.Perfection R p) p) f) = (coeff R p n) f
theorem Perfection.coeff_iterate_frobenius {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n m : ) :
(coeff R p (n + m)) ((⇑(frobenius (Ring.Perfection R p) p))^[m] f) = (coeff R p n) f
theorem Perfection.coeff_iterate_frobenius' {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] (f : Ring.Perfection R p) (n m : ) (hmn : m n) :
(coeff R p n) ((⇑(frobenius (Ring.Perfection R p) p))^[m] f) = (coeff R p (n - m)) f
theorem Perfection.coeff_add_ne_zero {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] {f : Ring.Perfection R p} {n : } (hfn : (coeff R p n) f 0) (k : ) :
(coeff R p (n + k)) f 0
theorem Perfection.coeff_ne_zero_of_le {R : Type u₁} [CommSemiring R] {p : } [hp : Fact (Nat.Prime p)] [CharP R p] {f : Ring.Perfection R p} {m n : } (hfm : (coeff R p m) f 0) (hmn : m n) :
(coeff R p n) f 0
instance Perfection.perfectRing (R : Type u₁) [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] :
noncomputable def Perfection.lift (p : ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] :

Given rings R and S of characteristic p, with R being perfect, any homomorphism R →+* S can be lifted to a homomorphism R →+* Perfection S p.

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  • One or more equations did not get rendered due to their size.
@[simp]
theorem Perfection.lift_symm_apply (p : ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (f : R →+* Ring.Perfection S p) :
(lift p R S).symm f = (coeff S p 0).comp f
@[simp]
theorem Perfection.lift_apply_apply_coe (p : ) [hp : Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (f : R →+* S) (r : R) (n : ) :
(((lift p R S) f) r) n = f ((⇑(frobeniusEquiv R p).symm)^[n] r)
theorem Perfection.hom_ext (p : ) [hp : Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {f g : R →+* Ring.Perfection S p} (hfg : ∀ (x : R), (coeff S p 0) (f x) = (coeff S p 0) (g x)) :
f = g
def Perfection.map {R : Type u₁} [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) :

A ring homomorphism R →+* S induces Perfection R p →+* Perfection S p.

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@[simp]
theorem Perfection.map_apply_coe {R : Type u₁} [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) (f : Ring.Perfection R p) (n : ) :
((map p φ) f) n = φ ((coeff R p n) f)
theorem Perfection.coeff_map {R : Type u₁} [CommSemiring R] (p : ) [hp : Fact (Nat.Prime p)] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) (f : Ring.Perfection R p) (n : ) :
(coeff S p n) ((map p φ) f) = φ ((coeff R p n) f)
structure PerfectionMap (p : ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₂} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* R) :

A perfection map to a ring of characteristic p is a map that is isomorphic to its perfection.

theorem PerfectionMap.mk' {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {f : P →+* R} (g : P ≃+* Ring.Perfection R p) (hfg : (Perfection.lift p P R) f = g) :

Create a PerfectionMap from an isomorphism to the perfection.

theorem PerfectionMap.of (p : ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] :

The canonical perfection map from the perfection of a ring.

theorem PerfectionMap.id (p : ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] :

For a perfect ring, it itself is the perfection.

noncomputable def PerfectionMap.equiv {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) :

A perfection map induces an isomorphism to the perfection.

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theorem PerfectionMap.equiv_apply {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) (x : P) :
m.equiv x = ((Perfection.lift p P R) π) x
theorem PerfectionMap.comp_equiv {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) (x : P) :
(Perfection.coeff R p 0) (m.equiv x) = π x
theorem PerfectionMap.comp_equiv' {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) :
(Perfection.coeff R p 0).comp m.equiv = π
theorem PerfectionMap.comp_symm_equiv {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) (f : Ring.Perfection R p) :
π (m.equiv.symm f) = (Perfection.coeff R p 0) f
theorem PerfectionMap.comp_symm_equiv' {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {π : P →+* R} (m : PerfectionMap p π) :
noncomputable def PerfectionMap.lift (p : ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) :
(R →+* S) (R →+* P)

Given rings R and S of characteristic p, with R being perfect, any homomorphism R →+* S can be lifted to a homomorphism R →+* P, where P is any perfection of S.

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@[simp]
theorem PerfectionMap.lift_symm_apply (p : ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) (f : R →+* P) :
(lift p R S P π m).symm f = π.comp f
@[simp]
theorem PerfectionMap.lift_apply (p : ) [Fact (Nat.Prime p)] (R : Type u₁) [CommSemiring R] [CharP R p] [PerfectRing R p] (S : Type u₂) [CommSemiring S] [CharP S p] (P : Type u₃) [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) (f : R →+* S) :
(lift p R S P π m) f = (↑m.equiv.symm).comp ((Perfection.lift p R S) f)
theorem PerfectionMap.hom_ext {p : } [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] [PerfectRing R p] {S : Type u₂} [CommSemiring S] [CharP S p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] (π : P →+* S) (m : PerfectionMap p π) {f g : R →+* P} (hfg : ∀ (x : R), π (f x) = π (g x)) :
f = g
noncomputable def PerfectionMap.map (p : ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} :
PerfectionMap p π{σ : Q →+* S} → (n : PerfectionMap p σ) → (φ : R →+* S) → P →+* Q

A ring homomorphism R →+* S induces P →+* Q, a map of the respective perfections.

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theorem PerfectionMap.comp_map (p : ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ) (φ : R →+* S) :
σ.comp (map p m n φ) = φ.comp π
theorem PerfectionMap.map_map (p : ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {P : Type u₃} [CommSemiring P] [CharP P p] [PerfectRing P p] {S : Type u₂} [CommSemiring S] [CharP S p] {Q : Type u₄} [CommSemiring Q] [CharP Q p] [PerfectRing Q p] {π : P →+* R} (m : PerfectionMap p π) {σ : Q →+* S} (n : PerfectionMap p σ) (φ : R →+* S) (x : P) :
σ ((map p m n φ) x) = φ (π x)
theorem PerfectionMap.map_eq_map (p : ) [Fact (Nat.Prime p)] {R : Type u₁} [CommSemiring R] [CharP R p] {S : Type u₂} [CommSemiring S] [CharP S p] (φ : R →+* S) :
map p φ = Perfection.map p φ
def ModP (O : Type u₂) [CommRing O] (p : ) :
Type u₂

O/(p) for O, ring of integers of K.

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instance ModP.commRing (O : Type u₂) [CommRing O] (p : ) :
Equations
instance ModP.charP (O : Type u₂) [CommRing O] (p : ) [Fact (Nat.Prime p)] [hvp : Fact ¬IsUnit p] :
CharP (ModP O p) p
instance ModP.nontrivial (O : Type u₂) [CommRing O] (p : ) [hp : Fact (Nat.Prime p)] [Fact ¬IsUnit p] :
noncomputable def ModP.preVal (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (p : ) (x : ModP O p) :

For a field K with valuation v : K → ℝ≥0 and ring of integers O, a function O/(p) → ℝ≥0 that sends 0 to 0 and x + (p) to v(x) as long as x ∉ (p).

Equations
theorem ModP.preVal_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {p : } :
preVal K v O p 0 = 0
theorem ModP.preVal_mk {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } {x : O} (hx : (Ideal.Quotient.mk (Ideal.span {p})) x 0) :
preVal K v O p ((Ideal.Quotient.mk (Ideal.span {p})) x) = v ((algebraMap O K) x)
theorem ModP.preVal_mul {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } {x y : ModP O p} (hxy0 : x * y 0) :
preVal K v O p (x * y) = preVal K v O p x * preVal K v O p y
theorem ModP.preVal_add {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } (x y : ModP O p) :
preVal K v O p (x + y) max (preVal K v O p x) (preVal K v O p y)
theorem ModP.v_p_lt_preVal {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } {x : ModP O p} :
v p < preVal K v O p x x 0
theorem ModP.preVal_eq_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } {x : ModP O p} :
preVal K v O p x = 0 x = 0
theorem ModP.v_p_lt_val {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } {x : O} :
v p < v ((algebraMap O K) x) (Ideal.Quotient.mk (Ideal.span {p})) x 0
theorem ModP.mul_ne_zero_of_pow_p_ne_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } [hp : Fact (Nat.Prime p)] {x y : ModP O p} (hx : x ^ p 0) (hy : y ^ p 0) :
x * y 0
def PreTilt (O : Type u₂) [CommRing O] (p : ) :
Type u₂

Perfection of O/(p) where O is the ring of integers of K.

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instance PreTilt.instCharP (O : Type u₂) [CommRing O] (p : ) [Fact (Nat.Prime p)] [Fact ¬IsUnit p] :
CharP (PreTilt O p) p
noncomputable def PreTilt.valAux (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (p : ) [Fact (Nat.Prime p)] [Fact ¬IsUnit p] (f : PreTilt O p) :

The valuation Perfection(O/(p)) → ℝ≥0 as a function. Given f ∈ Perfection(O/(p)), if f = 0 then output 0; otherwise output preVal(f(n))^(p^n) for any n such that f(n) ≠ 0.

Equations
theorem PreTilt.coeff_nat_find_add_ne_zero {O : Type u₂} [CommRing O] {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] {f : PreTilt O p} {h : ∃ (n : ), (Perfection.coeff (ModP O p) p n) f 0} (k : ) :
(Perfection.coeff (ModP O p) p (Nat.find h + k)) f 0
theorem PreTilt.valAux_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] :
valAux K v O p 0 = 0
theorem PreTilt.valAux_eq {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] {f : PreTilt O p} {n : } (hfn : (Perfection.coeff (ModP O p) p n) f 0) :
valAux K v O p f = ModP.preVal K v O p ((Perfection.coeff (ModP O p) p n) f) ^ p ^ n
theorem PreTilt.valAux_one {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] :
valAux K v O p 1 = 1
theorem PreTilt.valAux_mul {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] (f g : PreTilt O p) :
valAux K v O p (f * g) = valAux K v O p f * valAux K v O p g
theorem PreTilt.valAux_add {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] (f g : PreTilt O p) :
valAux K v O p (f + g) max (valAux K v O p f) (valAux K v O p g)
noncomputable def PreTilt.val (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ) [Fact (Nat.Prime p)] [Fact ¬IsUnit p] :

The valuation Perfection(O/(p)) → ℝ≥0. Given f ∈ Perfection(O/(p)), if f = 0 then output 0; otherwise output preVal(f(n))^(p^n) for any n such that f(n) ≠ 0.

Equations
theorem PreTilt.map_eq_zero {K : Type u₁} [Field K] {v : Valuation K NNReal} {O : Type u₂} [CommRing O] [Algebra O K] (hv : v.Integers O) {p : } [Fact (Nat.Prime p)] [Fact ¬IsUnit p] {f : PreTilt O p} :
(val K v O hv p) f = 0 f = 0
theorem PreTilt.isDomain (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ) [Fact (Nat.Prime p)] [Fact ¬IsUnit p] :
def Tilt (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ) [Fact (Nat.Prime p)] [hvp : Fact (v p 1)] :
Type u₂

The tilt of a field, as defined in Perfectoid Spaces by Peter Scholze, as in [scholze2011perfectoid]. Given a field K with valuation K → ℝ≥0 and ring of integers O, this is implemented as the fraction field of the perfection of O/(p).

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noncomputable instance Tilt.instField (K : Type u₁) [Field K] (v : Valuation K NNReal) (O : Type u₂) [CommRing O] [Algebra O K] (hv : v.Integers O) (p : ) [Fact (Nat.Prime p)] [hvp : Fact (v p 1)] :
Field (Tilt K v O hv p)
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