Multiplicative characters of finite rings and fields #
R' be a commutative rings.
A multiplicative character of
R with values in
R' is a morphism of
monoids from the multiplicative monoid of
R into that of
that sends non-units to zero.
We use the namespace
MulChar for the definitions and results.
Main results #
We define a multiplicative character to be quadratic if its values
-1, and we prove some properties of quadratic characters.
Finally, we show that the sum of all values of a nontrivial multiplicative
character vanishes; see
Definitions related to multiplicative characters #
Even though the intended use is when domain and target of the characters are commutative rings, we define them in the more general setting when the domain is a commutative monoid and the target is a commutative monoid with zero. (We need a zero in the target, since non-units are supposed to map to zero.)
In this setting, there is an equivalence between multiplicative characters
R → R' and group homomorphisms
Rˣ → R'ˣ, and the multiplicative characters
have a natural structure as a commutative group.
- toFun : R → R'
Define a structure for multiplicative characters.
A multiplicative character from a commutative monoid
R to a commutative monoid with zero
is a homomorphism of (multiplicative) monoids that sends non-units to zero.
This is the corresponding extension of
Equivalence of multiplicative characters with homomorphisms on units #
We show that restriction / extension by zero gives an equivalence
MulChar R R' and
Rˣ →* R'ˣ.
Commutative group structure on multiplicative characters #
The multiplicative characters
R → R' form a commutative group.
Properties of multiplicative characters #
We introduce the properties of being nontrivial or quadratic and prove some basic facts about them.
We now assume that domain and target are commutative rings.
If two values of quadratic characters with target
ℤ agree after coercion into a ring
of characteristic not
2, then they agree in
Composition with an injective ring homomorphism preserves nontriviality.
Composition with a ring homomorphism preserves the property of being a quadratic character.
pth power of a quadratic character is itself, when
p is the (prime) characteristic
of the target ring.
The sum over all values of a nontrivial multiplicative character on a finite ring is zero (when the target is a domain).
The sum over all values of the trivial multiplicative character on a finite ring is the cardinality of its unit group.