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/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import data.polynomial.cardinal
import ring_theory.algebraic
/-!
### Cardinality of algebraic numbers
In this file, we prove variants of the following result: the cardinality of algebraic numbers under
an R-algebra is at most `# polynomial R * ℵ₀`.
Although this can be used to prove that real or complex transcendental numbers exist, a more direct
proof is given by `liouville.is_transcendental`.
-/
universes u v
open cardinal polynomial
open_locale cardinal
namespace algebraic
theorem aleph_0_le_cardinal_mk_of_char_zero (R A : Type*) [comm_ring R] [is_domain R]
[ring A] [algebra R A] [char_zero A] : ℵ₀ ≤ #{x : A // is_algebraic R x} :=
@mk_le_of_injective (ulift ℕ) {x : A | is_algebraic R x} (λ n, ⟨_, is_algebraic_nat n.down⟩)
(λ m n hmn, by simpa using hmn)
section lift
variables (R : Type u) (A : Type v) [comm_ring R] [comm_ring A] [is_domain A] [algebra R A]
[no_zero_smul_divisors R A]
theorem cardinal_mk_lift_le_mul :
cardinal.lift.{u v} (#{x : A // is_algebraic R x}) ≤ cardinal.lift.{v u} (#(polynomial R)) * ℵ₀ :=
begin
rw [←mk_ulift, ←mk_ulift],
let g : ulift.{u} {x : A | is_algebraic R x} → ulift.{v} (polynomial R) :=
λ x, ulift.up (classical.some x.1.2),
apply cardinal.mk_le_mk_mul_of_mk_preimage_le g (λ f, _),
suffices : fintype (g ⁻¹' {f}),
{ exact @mk_le_aleph_0 _ (@fintype.to_encodable _ this) },
by_cases hf : f.1 = 0,
{ convert set.fintype_empty,
apply set.eq_empty_iff_forall_not_mem.2 (λ x hx, _),
simp only [set.mem_preimage, set.mem_singleton_iff] at hx,
apply_fun ulift.down at hx,
rw hf at hx,
exact (classical.some_spec x.1.2).1 hx },
let h : g ⁻¹' {f} → f.down.root_set A := λ x, ⟨x.1.1.1, (mem_root_set_iff hf x.1.1.1).2 begin
have key' : g x = f := x.2,
simp_rw ← key',
exact (classical.some_spec x.1.1.2).2
end⟩,
apply fintype.of_injective h (λ _ _ H, _),
simp only [subtype.val_eq_coe, subtype.mk_eq_mk] at H,
exact subtype.ext (ulift.down_injective (subtype.ext H))
end
theorem cardinal_mk_lift_le_max :
cardinal.lift.{u v} (#{x : A // is_algebraic R x}) ≤ max (cardinal.lift.{v u} (#R)) ℵ₀ :=
(cardinal_mk_lift_le_mul R A).trans $
(mul_le_mul_right' (lift_le.2 cardinal_mk_le_max) _).trans $ by simp [le_total]
theorem cardinal_mk_lift_le_of_infinite [infinite R] :
cardinal.lift.{u v} (#{x : A // is_algebraic R x}) ≤ cardinal.lift.{v u} (#R) :=
(cardinal_mk_lift_le_max R A).trans $ by simp
variable [encodable R]
@[simp] theorem countable_of_encodable : set.countable {x : A | is_algebraic R x} :=
begin
rw [←mk_set_le_aleph_0, ←lift_le],
apply (cardinal_mk_lift_le_max R A).trans,
simp
end
@[simp] theorem cardinal_mk_of_encodable_of_char_zero [char_zero A] [is_domain R] :
#{x : A // is_algebraic R x} = ℵ₀ :=
le_antisymm (by simp) (aleph_0_le_cardinal_mk_of_char_zero R A)
end lift
section non_lift
variables (R A : Type u) [comm_ring R] [comm_ring A] [is_domain A] [algebra R A]
[no_zero_smul_divisors R A]
theorem cardinal_mk_le_mul : #{x : A // is_algebraic R x} ≤ #(polynomial R) * ℵ₀ :=
by { rw [←lift_id (#_), ←lift_id (#(polynomial R))], exact cardinal_mk_lift_le_mul R A }
theorem cardinal_mk_le_max : #{x : A // is_algebraic R x} ≤ max (#R) ℵ₀ :=
by { rw [←lift_id (#_), ←lift_id (#R)], exact cardinal_mk_lift_le_max R A }
theorem cardinal_mk_le_of_infinite [infinite R] : #{x : A // is_algebraic R x} ≤ #R :=
(cardinal_mk_le_max R A).trans $ by simp
end non_lift
end algebraic
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