Further Results on multiplicative characters #
Two multiplicative characters on a monoid whose unit group is generated by g
are equal if and only if they agree on g
.
The values of a multiplicative character on R
are n
th roots of unity, where n = #Rˣ
.
Multiplicative characters on finite monoids with cyclic unit group #
The order of the unit group of a finite monoid as a PNat
(for use in rootsOfUnity
).
Equations
- MulChar.Monoid.orderUnits M = ⟨Fintype.card Mˣ, ⋯⟩
Instances For
Given a finite monoid M
with unit group Mˣ
cyclic of order n
and an n
th root of
unity ζ
in R
, there is a multiplicative character M → R
that sends a given generator
of Mˣ
to ζ
.
Equations
Instances For
The group of multiplicative characters on a finite monoid M
with cyclic unit group Mˣ
of order n
is isomorphic to the group of n
th roots of unity in the target R
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Multiplicative characters on finite fields #
There is a character of order n
on F
if #F ≡ 1 mod n
and the target contains
a primitive n
th root of unity.
There is always a character on F
of order #F-1
with values in a ring that has
a primitive (#F-1)
th root of unity.
The non-zero values of a multiplicative character χ
such that χ^n = 1
are n
th roots of unity.
If χ
is a multiplicative character with χ^n = 1
and μ
is a primitive n
th root
of unity, then, for a ≠ 0
, there is some k
such that χ a = μ^k
.
The values of a multiplicative character χ
such that χ^n = 1
are contained in ℤ[μ]
when
μ
is a primitive n
th root of unity.
The values of a multiplicative character of order n
are contained in ℤ[μ]
when
μ
is a primitive n
th root of unity.