Dependencies

The Hurwitz quaternions are the set \(\mathcal{O}:= \mathbb {Z}\oplus \mathbb {Z}\omega \oplus \mathbb {Z}i\oplus \mathbb {Z}i\omega \) (as an abstract abelian group or as a subgroup of the usual quaternions). Here \(\omega =\frac{-1+(i+j+k)}{2}\) and note that \((i+j+k)^2=-3\). We have \(\overline{\omega }=\omega ^2=-(\omega +1)\). A general quaternion \(a+bi+cj+dk\) is a Hurwitz quaternion if either \(a,b,c,d\in \mathbb {Z}\) or \(a,b,c,d\in \mathbb {Z}+\frac{1}{2}\).

Every element of \(\widehat{D}\) can be written as \(z/N\) with \(z\in \widehat{\mathcal{O}}\) and \(N\in \mathbb {N}^+\).

The group of units of \(\widehat{D}\) is \(D^\times \widehat{\mathcal{O}}^\times \). More precisely, every element of \(\widehat{D}^\times \) can be written as a product \(\delta u\) with \(\delta \in D^\times \) and \(u\in \widehat{\mathcal{O}}^\times \).

Given a “usual” quaternion \(a=x+yi+zj+wk\) with \(x,y,z,w\in \mathbb {R}\), there exists a Hurwitz quaternion \(q\) such that \(N(a-q){\lt}1\).

All left ideals of \(\mathcal{O}\) are principal.

The Hurwitz quaternions come equipped with an integer-valued norm, which is \(a^2+b^2+c^2+d^2\) on \(a+bi+cj+dk\) but needs to be modified a bit to deal with \(\omega \).

We have \(N(x)=x\overline{x}\).

The norm of an element is zero if and only if the element is zero.

The norm of a product is the product of the norms.

The norm of an element is nonnegative.

The norm of \(1\) is \(1\).

The norm of \(0\) is \(0\).

Given two Hurwitz quaternions \(a\) and \(b\) with \(b\) nonzero, there exists \(q\) and \(r\) such that \(a=qb+r\) and \(N(r){\lt}N(b)\).

The Hurwitz quaternions form a ring.

There’s a conjugation map (which we’ll call "star") from the Hurwitz quaternions to themselves, sending integers to themselves and purely imaginary elements like \(2\omega +1\) to minus themselves. It satisfies \((x^*)^*=x\), \((xy)^*=y^*x^*\) and \((x+y)^*=x^*+y^*\). In particular, the Hurwitz quaternions are a "star ring" in the sense of mathlib.

If \(N\) is a positive natural then the obvious map \(\mathcal{O}\to \widehat{\mathcal{O}}/N\widehat{\mathcal{O}}\) is surjective.

The profinite completion \(\widehat{\mathbb {Q}}\) of \(\mathbb {Q}\) is the tensor product \(\mathbb {Q}\otimes _{\mathbb {Z}}\widehat{\mathbb {Z}}\), or \(\widehat{\mathbb {Q}}=\mathbb {Q}\otimes \widehat{\mathbb {Z}}\) for short.

Every element of \(\widehat{\mathbb {Q}}:=\mathbb {Q}\otimes \widehat{\mathbb {Z}}\) can be written as \(q\otimes _t z\) with \(q\in \mathbb {Q}\) and \(z\in \widehat{\mathbb {Z}}\). Furthermore one can even assume that \(q=\frac{1}{N}\) for some positive integer \(N\).

The ring homomorphism \(\mathbb {Q}\to \widehat{\mathbb {Q}}\) sending \(q\) to \(q\otimes _t 1\) is injective.

The ring homomorphism \(\widehat{\mathbb {Z}}\to \widehat{\mathbb {Q}}\) sending \(z\) to \(1\otimes _t z\) is injective.

If \(N\in \mathbb {N}^+\) and \(z\in \widehat{\mathbb {Z}}\) then we say that \(N\) and \(z\) are coprime if \(z_N\in (\mathbb {Z}/N\mathbb {Z})^\times \). We write \(z/N\) as notation for the element \(\frac{1}{N}\otimes _tz\).

Every element of \(\widehat{\mathbb {Q}}\) can be uniquely written as \(z/N\) with \(z\in \widehat{\mathbb {Z}}\), \(N\in \mathbb {N}^+\), and with \(N\) and \(z\) coprime.

The sum of \(\mathbb {Q}\) and \(\widehat{\mathbb {Z}}\) in \(\widehat{\mathbb {Q}}\) is \(\widehat{\mathbb {Q}}\). More precisely, every element of \(\widehat{\mathbb {Q}}\) can be written as \(q+z\) with \(q\in \mathbb {Q}\) and \(z\in \widehat{\mathbb {Z}}\), or more precisely as \(q\otimes _t 1+1\otimes _t z\).

The intersection of \(\mathbb {Q}\) and \(\widehat{\mathbb {Z}}\) in \(\widehat{\mathbb {Q}}\) is \(\mathbb {Z}\).

The product of \(\mathbb {Q}^\times \) and \(\widehat{\mathbb {Z}}^\times \) in \(\widehat{\mathbb {Q}}^\times \) is all of \(\widehat{\mathbb {Q}}^\times \). More precisely, every element of \(\widehat{\mathbb {Q}}^\times \) can be written as \(qz\) with \(q\in \mathbb {Q}^\times \) and \(z\in \widehat{\mathbb {Z}}^\times \).

The intersection of \(\mathbb {Q}^\times \) and \(\widehat{\mathbb {Z}}^\times \) in \(\widehat{\mathbb {Q}}^\times \) is \(\mathbb {Z}^\times \).

The profinite completion \(\widehat{\mathbb {Z}}\) of \(\mathbb {Z}\) is the set of all compatible collections \(c=(c_N)_N\) of elements of \(\mathbb {Z}/N\mathbb {Z}\) indexed by \(\mathbb {N}^+:=\{ 1,2,3,\ldots \} \). A collection is said to be compatible if for all positive integers \(D\mid N\), we have \(c_N\) mod \(D\) equals \(c_D\).

The map from the naturals into \(\widehat{\mathbb {Z}}\) sending \(n\) to \(n\) is injective.

\(\widehat{\mathbb {Z}}\) is a subring of \(\prod _{N\geq 1}(Z/N\mathbb {Z})\) and in particular is a ring.

The infinite sum \(0!+1!+2!+3!+4!+5!+\cdots \) looks like it makes no sense at all; it is the sum of an infinite series of larger and larger positive numbers. However, the sum is finite modulo \(N\) for every positive integer \(N\), because all the terms from \(N!\) onwards are multiples of \(N\) and thus are zero in \(\mathbb {Z}/N\mathbb {Z}\). Thus it makes sense to define \(e_N\) to be the value of the finite sum modulo \(N\). Explicitly, \(e_N=0!+1!+\cdots +(N-1)!\) modulo \(N\).

The collection \((e_N)_N\) is an element of \(\widehat{\mathbb {Z}}\).

The element \((e_N)_N\) of \(\widehat{\mathbb {Z}}\) is not in \(\mathbb {Z}\).

The multiples of \(N\) in \(\widehat{\mathbb {Z}}\) are precisely the compatible collections \((z_i)_i\in \widehat{\mathbb {Z}}\) with \(z_N=0\).

\(0\not=1\) in \(\widehat{\mathbb {Z}}\).

If \(0{\lt}N\) is an integer then multiplication by \(N\) is injective on \(\widehat{\mathbb {Z}}\).