Theorems · Theorem · number theory
MulChar.card_eq_card_units_of_hasEnoughRootsOfUnity
∀ (M : Type u_1) (R : Type u_2) [inst : CommMonoid M] [inst_1 : CommRing R] [Finite M] [HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)], Nat.card (MulChar M R) = Nat.card Mˣ
The cardinality of the group of R-valued multiplicative characters on a finite commutative
monoid M is the same as that of its unit group Mˣ when R is a ring that has enough roots
of unity.
- Defined in
- Mathlib.NumberTheory.MulChar.Duality
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Finitestatement and proof · cited by 3,029
- Unitsstatement and proof · cited by 2,804
- CommMonoidstatement and proof · cited by 2,264
- Nat.cardstatement · cited by 844
- Nonempty.someproof · cited by 340
- MulCharstatement · cited by 186
- Nat.card_congrproof · cited by 133
- Monoid.exponentstatement and proof · cited by 128
- MulEquiv.toEquivproof · cited by 126
- HasEnoughRootsOfUnitystatement and proof · cited by 56
- MulChar.mulEquiv_unitsproof · cited by 2
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