Lemmas of Gauss and Eisenstein #
This file contains the Lemmas of Gauss and Eisenstein on the Legendre symbol.
The main results are ZMod.gauss_lemma and ZMod.eisenstein_lemma.
theorem
ZMod.Ico_map_valMinAbs_natAbs_eq_Ico_map_id
(p : ℕ)
[hp : Fact (Nat.Prime p)]
(a : ZMod p)
(hap : a ≠ 0)
:
Multiset.map (fun (x : ℕ) => (a * ↑x).valMinAbs.natAbs) (Finset.Ico 1 (p / 2).succ).val = Multiset.map (fun (a : ℕ) => a) (Finset.Ico 1 (p / 2).succ).val
The image of the map sending a nonzero natural number x ≤ p / 2 to the absolute value
of the integer in (-p/2, p/2] that is congruent to a * x mod p is the set
of nonzero natural numbers x such that x ≤ p / 2.
theorem
ZMod.gauss_lemma
{p : ℕ}
[h : Fact (Nat.Prime p)]
{a : ℤ}
(hp : p ≠ 2)
(ha0 : ↑a ≠ 0)
:
legendreSym p a = (-1) ^ (Finset.filter (fun (x : ℕ) => p / 2 < (↑a * ↑x).val) (Finset.Ico 1 (p / 2).succ)).card
Gauss' lemma. The Legendre symbol can be computed by considering the number of naturals less
than p/2 such that (a * x) % p > p / 2.
theorem
ZMod.div_eq_filter_card
{a : ℕ}
{b : ℕ}
{c : ℕ}
(hb0 : 0 < b)
(hc : a / b ≤ c)
:
a / b = (Finset.filter (fun (x : ℕ) => x * b ≤ a) (Finset.Ico 1 c.succ)).card
theorem
ZMod.sum_mul_div_add_sum_mul_div_eq_mul
(p : ℕ)
(q : ℕ)
[hp : Fact (Nat.Prime p)]
(hq0 : ↑q ≠ 0)
:
Each of the sums in this lemma is the cardinality of the set of integer points in each of the
two triangles formed by the diagonal of the rectangle (0, p/2) × (0, q/2). Adding them
gives the number of points in the rectangle.