Rational Wave Numbers and the Algebraic Structure of the Cyclic Groups of the Roots of Unity
Terence R. Smith
TL;DR
The paper develops a wave-based algebraic framework that generalizes the cyclic groups of the $n$th roots of unity by constructing three nested structures: a multiplicative group ${\bf W}_M$, a additive ring ${\bf W}_A$, and a field ${\bf W}_I$, all generated from two unitary wave numbers. Each element has a polar form ${\boldsymbol\omega}={\bf A}{\bf w}(f,g)$ with ${\bf w}(f,g)=e^{2\pi i( f {\boldsymbol\xi} \oplus g {\bf 1})}$ and rational parameters $f,g$, while amplitudes ${\bf A}$ are products of trigonometric functions; closures under ${\otimes}$, ${\oplus}$, and ${\iota}$ yield a rich arithmetic, including sums and differences expressed via cosine/sine of logarithmic ratios. The work develops orthonormal bases for additive and invertible wave numbers, defines integral, particulate, and prime wave numbers, and constructs a prime-detection mechanism using co- and re-numbers with circular products, embedding number-theoretic ideas into a wave-theoretic formalism. These structures provide a wave-centered perspective on numbers and offer potential connections to wave equations and spectral representations beyond classical roots-of-unity theory.
Abstract
Rational wave numbers are periodic sequences ${\mathbf ω}={\bf A}{\bf w}(f,g)$ in which amplitude ${\bf A}$ a product of powers of trigonometric sequences and ${\bf w}(f,g)=\exp({\bf {i2}π( f {\mathbf ξ} \oplus g{\bf 1})})$ is a sequence with $ ξε\mathbb{ Z}$ and $f,g$ rational. They generalize the cyclic groups of the $n$th roots of unity and are generated from two unitary sequences. The multiplicative group ${\bf W}_M$ with ${\bf A}={\bf 1}$ is their closure wrt product, root, and reflection operators. The commutative ring ${\bf W}_A$ has additional closure wrt summation. The field ${\bf W}_I$ of invertible wave numbers has further closure wrt an inverse. Sums and differences of its elements are ${\mathbf ω}_1 \oplus {\mathbf ω}_2= {\bf 2} \cos\big( {\bf i} ln\big(\frac{{\mathbf ω}_2}{{\mathbf ω}_1}\big)^{1/2}\big) \big({\mathbf ω}_1 {\mathbf ω}_2 \big)^{1/2}$ and ${\mathbf ω}_1 \ominus {\mathbf ω}_2= {\bf {2 i}} \sin\big( {\bf i} ln\big(\frac{{\mathbf ω}_2}{{\mathbf ω}_1}\big)^{1/2}\big) \big({\mathbf ω}_1 {\mathbf ω}_2 \big)^{1/2}$. Its amplitudes form a multiplicative subgroup over which ${\bf W}_M \cup \big\{{\bf 0}\big\}$ is a vector space. Wave numbers of period $n\ ε \mathbb{N}$ possess $n$ phases, multiplicative norms, representations wrt orthonormal bases, and prime representations. Rational wave numbers may be completed with respect to Cauchy sequences of parameters $({f,g})$. Equations in invertible wave numbers have solutions corresponding to zeros of their trigonometric factors. Orthonormal bases are employed in constructing the integral wave numbers and allow definitions of the particulate and prime wave numbers in terms of permissible phase-values.