dedekind::numbers::UnitRoot< N > Struct Template Reference

A SYMBOLIC N-th root of unity \(\zeta_N^{\,p}\), carried by its exponent p ∈ ℤ/N (Modular<N>). The unit harmonics \(\mu_N=\{z:z^N=1\}\): the torsion subgroup of the unit circle U(1), cut out of ℂ, cyclic of order N and \(\cong\mathbb{Z}/N\) via de Moivre. More...

Public Member Functions
constexpr UnitRoot	inverse () const
	The inverse \(\zeta^{-p}\) (= conjugate, on the unit circle).
constexpr Complex< QuadraticReal< 2 > >	resolve () const
	Collapse to the exact ℂ(ℚ(√2)) value via de Moivre — only N=8, where \(\zeta_8\in\mathbb{Q}(\sqrt2)\); off that base this member is unavailable (a compile-time Honest Rejection).

Public Attributes
dedekind::morphologies::Modular< N >	exponent {}

Friends
constexpr bool	operator== (const UnitRoot &, const UnitRoot &)=default
constexpr UnitRoot	operator* (const UnitRoot &a, const UnitRoot &b)
	The μ_N group law: harmonic product = exponent sum (de Moivre).

Detailed Description

template<unsigned N>
struct dedekind::numbers::UnitRoot< N >

A SYMBOLIC N-th root of unity \(\zeta_N^{\,p}\), carried by its exponent p ∈ ℤ/N (Modular<N>). The unit harmonics \(\mu_N=\{z:z^N=1\}\): the torsion subgroup of the unit circle U(1), cut out of ℂ, cyclic of order N and \(\cong\mathbb{Z}/N\) via de Moivre.

Arithmetic is multiplicative and symbolic — \(\zeta^p \cdot \zeta^q = \zeta^{p+q}\), exponent addition in ℤ/N, never a numeric ℂ product. A group, NOT a ring: a sum of harmonics generally leaves \(\mu_N\) (that sum is the ℂ-linear span — the function-space semimodule — not the harmonics). Collapse to an exact ℂ value is deferred to resolve() and exact only where \(\zeta_N\) lives in the base field: N=8 ⊂ ℚ(√2), via root8. Other N would need the cyclotomic field \(\mathbb{Q}(\zeta_N)\) and get no numeric collapse here (a compile-time Honest Rejection).

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/numbers/complex.cppm