FundamentalDiscriminant.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

Fundamental discriminants

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

References:

Wikipedia, Fundamental discriminant

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def NumberField.IsFundamentalDiscr Fundamental discriminants are those integers D that appear as discriminants of quadratic fields line 39 source
Source documentation
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.

standard axioms: propext what this means
Statement uses: Int
Proof uses: Or, And, Dvd.dvd, OfNat.ofNat, Not, Int.ModEq, HDiv.hDiv, Squarefree, Ne
5 instances, coercions and auxiliary constants left out.
```
def IsFundamentalDiscr (D : ℤ) : Prop :=
…
```
theorem NumberField.isFundamentalDiscr_iff Documented declaration: isFundamentalDiscr_iff line 45 source
Source documentation
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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Int, Iff, NumberField.IsFundamentalDiscr, Or, Exists, And, Eq, HMul.hMul, OfNat.ofNat, HMod.hMod, Squarefree, Ne
Proof uses: Not, Dvd.dvd, False, absurd, IsUnit, Exists.intro, Decidable.byContradiction, of_eq_true, Eq.trans, True, congrArg, Neg.neg, congr, Nat, eq_false', False.elim, noConfusion_of_Nat, Int.ctorIdx, or_self, not_false_eq_true and 23 more
40 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.
```
theorem isFundamentalDiscr_iff {D : ℤ} :
    IsFundamentalDiscr D ↔
      (∃ m, D = 4 * m ∧ (m % 4 = 2 ∨ m % 4 = 3) ∧ Squarefree m) ∨
        D ≠ 1 ∧ D % 4 = 1 ∧ Squarefree D := by
…
```
Mentions: NumberField.IsFundamentalDiscr