Documentation

Mathlib.Data.ZMod.Units

Lemmas about units in ZMod. #

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def ZMod.unitsMap {n m : ℕ} (hm : n ∣ m) :
(ZMod m)ˣ →* (ZMod n)ˣ

unitsMap is a group homomorphism that maps units of ZMod m to units of ZMod n when n divides m.

Equations
Instances For
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    theorem ZMod.unitsMap_def {n m : ℕ} (hm : n ∣ m) :
    unitsMap hm = Units.map ↑(castHom hm (ZMod n))
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    theorem ZMod.unitsMap_comp {n m d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :
    (unitsMap hm).comp (unitsMap hd) = unitsMap ⋯
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    @[simp]
    theorem ZMod.unitsMap_self (n : ℕ) :
    unitsMap ⋯ = MonoidHom.id (ZMod n)ˣ
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    theorem ZMod.unitsMap_val {n m : ℕ} (h : n ∣ m) (a : (ZMod m)ˣ) :
    ↑((unitsMap h) a) = (↑a).cast

    unitsMap_val shows that coercing from (ZMod m)ˣ to ZMod n gives the same result when going via (ZMod n)ˣ and ZMod m.

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    theorem ZMod.isUnit_cast_of_dvd {n m : ℕ} (hm : n ∣ m) (a : (ZMod m)ˣ) :
    IsUnit (↑a).cast
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    @[deprecated ZMod.isUnit_cast_of_dvd (since := "2024-12-16")]
    theorem ZMod.IsUnit_cast_of_dvd {n m : ℕ} (hm : n ∣ m) (a : (ZMod m)ˣ) :
    IsUnit (↑a).cast

    Alias of ZMod.isUnit_cast_of_dvd.

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    theorem ZMod.unitsMap_surjective {n m : ℕ} [hm : NeZero m] (h : n ∣ m) :
    Function.Surjective ⇑(unitsMap h)
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    theorem ZMod.not_isUnit_of_mem_primeFactors {n p : ℕ} (h : p ∈ n.primeFactors) :
    ¬IsUnit ↑p
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    theorem ZMod.eq_unit_mul_divisor {N : ℕ} (a : ZMod N) :
    ∃ (d : ℕ), d ∣ N ∧ ∃ (u : ZMod N), IsUnit u ∧ a = u * ↑d

    Any element of ZMod N has the form u * d where u is a unit and d is a divisor of N.