1. FormalConjecturesForMathlib
  2. NumberTheory
  3. NumberField
  4. FundamentalDiscriminant
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Fundamental discriminants

A fundamental discriminant is an integer that occurs as the discriminant of a quadratic field.

References:

namespace NumberField

Fundamental discriminants are those integers D that appear as discriminants of quadratic fields.

D is a fundamental discriminant if it is either of the form 4m for m congruent to 2 or 3 mod 4 squarefree, or if it congruent to 1 mod 4 and squarefree.

def IsFundamentalDiscr (D : ℤ) : Prop := 4 ∣ D ∧ ¬ D / 4 ≡ 1 [ZMOD 4] ∧ Squarefree (D / 4) ∨ D ≠ 1 ∧ D ≡ 1 [ZMOD 4] ∧ Squarefree D

Unfolding of IsFundamentalDiscr in terms of %: a fundamental discriminant is either 4 times a squarefree integer congruent to 2 or 3 mod 4, or a squarefree integer congruent to 1 mod 4 other than 1 itself.

hsq:∀ {m : ℤ}, Squarefree m → ¬4 ∣ mm:ℤhm₄:m % 4 = 2 ∨ m % 4 = 3hm:Squarefree m⊢ 4 ∣ 4 * m ∧ ¬m % 4 = 1 ∧ Squarefree m ∨ ¬4 * m = 1 ∧ 4 * m % 4 = 1 ∧ Squarefree (4 * m) exact .inl ⟨⟨m, rfl⟩, hsq:∀ {m : ℤ}, Squarefree m → ¬4 ∣ mm:ℤhm₄:m % 4 = 2 ∨ m % 4 = 3hm:Squarefree m⊢ ¬m % 4 = 1 All goals completed! 🐙, hm⟩ D:ℤhsq:∀ {m : ℤ}, Squarefree m → ¬4 ∣ mh:¬D = 1 ∧ D % 4 = 1 ∧ Squarefree D⊢ 4 ∣ D ∧ ¬D / 4 % 4 = 1 ∧ Squarefree (D / 4) ∨ ¬D = 1 ∧ D % 4 = 1 ∧ Squarefree D All goals completed! 🐙end NumberField