Are roots of unity in hypercomplex algebras well defined?

Question

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the quaternions, octonions, and other hypercomplex algebras? Are there higher dimensional cyclotomic fields, and if so do they have any interesting properties like unique factorization or interesting morphisms?

Answer 1

Are the roots of unity well defined in the quaternions, octonions, and other hypercomplex algebras?

They can still be defined as elements satisfying $x^n=1$ for a nonnegative integer $n$. (I'm not sure what other definition you would use... )

Of course, that doesn't mean that they'll behave as nicely as those roots of unity in the complex plane. I'll use the quaternions $\Bbb H$ to illustrate.

For one thing, there are uncountably many $n$th roots of unity. In the complex plane, $(\cos(2\pi/n)+i\sin(2\pi/n))^n=1$ owing to de Moivre's identity. It turns out that in this formula you can replace $i$ with any quaternion $u$ which has zero real part and which satisfies $u^2=-1$, and there are uncountably many choices for such a $u$. Basically, each choice of $u$ picks out a copy of $\Bbb C$ sitting inside $\Bbb H$.

Are there higher dimensional cyclotomic fields, and if so do they have any interesting properties like unique factorization or interesting morphisms?

By "higher dimensional" and the first question, it looks like want to do something like adjoin roots of unity to $\Bbb H$, or some larger extension. Of course, such an extension would no longer be a field since it would lose commutativity, (and also associativity if you tried the octonions.)

Losing commutativity would force you to rethink what you wanted from factorizations. It is no longer possible to freely rearrange the order of factors, for example.

Answer 2

In tessarines they are well defined. In 4-dimensional tessarines a polynomial of the order $n$ has $n^2$ roots (accounting for root multiplicity). This is called fundamental theorem of tessarine algebra.