Ostrowski’s Theorem

The goal of this file is to prove Ostrowski’s Theorem which gives a list of all the nontrivial absolute values on a number field up to equivalence. (TODO)

References

• [K. Conrad, Ostroski's Theorem for Q][conradQ]
• [K. Conrad, Ostroski for number fields][conradnumbfield]
• [J. W. S. Cassels, Local fields][cassels1986local]

Tags

ring_norm, ostrowski

theorem Rat.MulRingNorm.eq_on_nat_iff_eq_on_Int {f : MulRingNorm ℚ} {g : MulRingNorm ℚ} : (∀ (n : ℕ), f ↑n = g ↑n) ↔ ∀ (n : ℤ), f ↑n = g ↑n

Values of a multiplicative norm of the rationals coincide on ℕ if and only if they coincide on ℤ.

theorem Rat.MulRingNorm.eq_on_nat_iff_eq {f : MulRingNorm ℚ} {g : MulRingNorm ℚ} : (∀ (n : ℕ), f ↑n = g ↑n) ↔ f = g

Values of a multiplicative norm of the rationals are determined by the values on the natural numbers.

theorem Rat.MulRingNorm.equiv_on_nat_iff_equiv {f : MulRingNorm ℚ} {g : MulRingNorm ℚ} : (∃ (c : ℝ), 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n) ↔ f.equiv g

The equivalence class of a multiplicative norm on the rationals is determined by its values on the natural numbers.

def Rat.MulRingNorm.mulRingNorm_padic (p : ℕ) [Fact (Nat.Prime p)] : MulRingNorm ℚ

The mulRingNorm corresponding to the p-adic norm on ℚ.

Equations

• Rat.MulRingNorm.mulRingNorm_padic p = { toFun := fun (x : ℚ) => ↑(padicNorm p x), map_zero' := ⋯, add_le' := ⋯, neg' := ⋯, map_one' := ⋯, map_mul' := ⋯, eq_zero_of_map_eq_zero' := ⋯ }

@[simp]

theorem Rat.MulRingNorm.mulRingNorm_eq_padic_norm (p : ℕ) [Fact (Nat.Prime p)] (r : ℚ) : (Rat.MulRingNorm.mulRingNorm_padic p) r = ↑(padicNorm p r)

theorem Rat.MulRingNorm.exists_minimal_nat_zero_lt_mulRingNorm_lt_one {f : MulRingNorm ℚ} (hf_nontriv : f ≠ 1) (bdd : ∀ (n : ℕ), f ↑n ≤ 1) : ∃ (p : ℕ), (0 < f ↑p ∧ f ↑p < 1) ∧ ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m

There exists a minimal positive integer with absolute value smaller than 1.

theorem Rat.MulRingNorm.is_prime_of_minimal_nat_zero_lt_mulRingNorm_lt_one {f : MulRingNorm ℚ} {p : ℕ} (hp0 : 0 < f ↑p) (hp1 : f ↑p < 1) (hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m) : Nat.Prime p

The minimal positive integer with absolute value smaller than 1 is a prime number.

theorem Rat.MulRingNorm.mulRingNorm_eq_one_of_not_dvd {f : MulRingNorm ℚ} (bdd : ∀ (n : ℕ), f ↑n ≤ 1) {p : ℕ} (hp0 : 0 < f ↑p) (hp1 : f ↑p < 1) (hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m) {m : ℕ} (hpm : ¬p ∣ m) : f ↑m = 1

A natural number not divible by p has absolute value 1.

theorem Rat.MulRingNorm.exists_pos_mulRingNorm_eq_pow_neg {f : MulRingNorm ℚ} {p : ℕ} (hp0 : 0 < f ↑p) (hp1 : f ↑p < 1) (hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m) : ∃ (t : ℝ), 0 < t ∧ f ↑p = ↑p ^ (-t)

The absolute value of p is p ^ (-t) for some positive real number t.

theorem Rat.MulRingNorm.mulRingNorm_equiv_padic_of_bounded {f : MulRingNorm ℚ} (hf_nontriv : f ≠ 1) (bdd : ∀ (n : ℕ), f ↑n ≤ 1) : ∃! p : ℕ, ∃ (hp : Fact (Nat.Prime p)), f.equiv (Rat.MulRingNorm.mulRingNorm_padic p)

If f is bounded and not trivial, then it is equivalent to a p-adic absolute value.