The cyclotomic character #
Let L be an integral domain and let n : ℕ+ be a positive integer. If μₙ is the
group of nth roots of unity in L then any field automorphism g of L
induces an automorphism of μₙ which, being a cyclic group, must be of
the form ζ ↦ ζ^j for some integer j = j(g), well-defined in ZMod d, with
d the cardinality of μₙ. The function j is a group homomorphism
(L ≃+* L) →* ZMod d.
Future work: If L is separably closed (e.g. algebraically closed) and p is a prime
number such that p ≠ 0 in L, then applying the above construction with
n = p^i (noting that the size of μₙ is p^i) gives a compatible collection of
group homomorphisms (L ≃+* L) →* ZMod (p^i) which glue to give
a group homomorphism (L ≃+* L) →* ℤₚ; this is the p-adic cyclotomic character.
Important definitions #
Let L be an integral domain, g : L ≃+* L and n : ℕ+. Let d be the number of nth roots
of 1 in L.
modularCyclotomicCharacter L n hn : (L ≃+* L) →* (ZMod n)ˣsendsgto the uniquejsuch thatg(ζ)=ζ^jfor allζ : rootsOfUnity n L. Herehnis a proof that there arennth roots of unity inL.cyclotomicCharacter L p : (L ≃+* L) →* ℤ_[p]ˣsendsgto the uniquejsuch thatg(ζ) = ζ ^ (j mod pⁱ)for allpⁱ-th roots of unityζ.Note: This is defined to be the trivial character if
Ldoes not have enough roots of unity.
Implementation note #
In theory this could be set up as some theory about monoids, being a character
on monoid isomorphisms, but under the hypotheses that the n-th roots of unity
are cyclic. The advantage of sticking to integral domains is that finite subgroups
are guaranteed to be cyclic, so the weaker assumption that there are n nth
roots of unity is enough. All the applications I'm aware of are when L is a
field anyway.
Although I don't know whether it's of any use, modularCyclotomicCharacter'
is the general case for integral domains, with target in (ZMod d)ˣ
where d is the number of nth roots of unity in L.
TODO #
- Prove the compatibility of
modularCyclotomicCharacter nandmodularCyclotomicCharacter mifn ∣ m.
Tags #
cyclotomic character
modularCyclotomicCharacter_aux g n is a non-canonical auxiliary integer j,
only well-defined modulo the number of n-th roots of unity in L, such that g(ζ)=ζ^j
for all n-th roots of unity ζ in L.
Instances For
If g is a ring automorphism of L, and n : ℕ+, then
modularCyclotomicCharacter.toFun n g is the j : ZMod d such that g(ζ)=ζ^j for all
n-th roots of unity. Here d is the number of nth roots of unity in L.
Instances For
The formula which characterises the output of modularCyclotomicCharacter g n.
If g(t)=t^c for all roots of unity, then c=χ(g).
Given a positive integer n, modularCyclotomicCharacter' n is a
multiplicative homomorphism from the automorphisms of a field L to (ℤ/dℤ)ˣ,
where d is the number of n-th roots of unity in L. It is uniquely
characterised by the property that g(ζ)=ζ^(modularCyclotomicCharacter n g)
for g an automorphism of L and ζ an nth root of unity.
Instances For
Given a positive integer n and a field L containing n nth roots
of unity, modularCyclotomicCharacter n is a multiplicative homomorphism from the
automorphisms of L to (ℤ/nℤ)ˣ. It is uniquely characterised by the property that
g(ζ)=ζ^(modularCyclotomicCharacter n g) for g an automorphism of L and ζ any nth root
of unity.
Instances For
The relationship between IsPrimitiveRoot.autToPow and
modularCyclotomicCharacter. Note that IsPrimitiveRoot.autToPow
needs an explicit root of unity, and also an auxiliary "base ring" R.
Suppose L is a domain containing all pⁱ-th primitive roots with p a (rational) prime.
If g is a ring automorphism of L, then cyclotomicCharacter L p g is the unique j : ℤₚ such
that g(ζ) = ζ ^ (j mod pⁱ) for all pⁱ-th roots of unity ζ.
Note: This is the trivial character when L does not contain all pⁱ-th primitive roots.