Dyadic cyclotomic ring for Clifford+T synthesis

Set $\omega =e^{i\pi \mathrm{/}4}$, $K=\mathbb{Q}(\omega )$, $O=\mathbb{Z}[\omega ]$, and $D=O[1\mathrm{/}2]=\mathbb{Z}[\omega ,1\mathrm{/}2]$. The relation ${\omega }^4=-1$ reduces every element of $O$ to four integer coefficients in the power basis $1,\omega ,{\omega }^2,{\omega }^3$. The canonical dyadic encoding of a nonzero $z\in D$ is

$z=(a_0+a_1\omega +a_2{\omega }^2+a_3{\omega }^3)\mathrm{/}{2}^r,$

where $r\ge 0$ is least and, if $r>0$, the four integers are not all even. Zero is encoded with $r=0$ and all four coefficients zero. Arithmetic is exact: multiply polynomials, reduce with ${\omega }^4=-1$, combine denominators, and remove the common power of $2$ from the coefficients. The integral-basis supplier Ring of integers of every cyclotomic field and the proof of Canonical dyadic coefficient encoding ↗ justify existence and uniqueness.

The ring $D$ is a subring of $K\subset \mathbb{C}$. All coefficients and equality tests in this page use the displayed power basis; no floating-point comparison is implicit.