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Solving equations over the hyperbolic numbers is weird; for example, the equation $x^2 = 1$ has four solutions; not only are $1$ and $-1$ solutions, but so too are $j$ and $-j$.

Question. Are there any good resources out there for learning how to solve polynomial equations over the hyperbolic numbers? I'm especially interested in the problem of finding all $n$th roots of unity.

Remark. The familiar formula $$x^2 + bx = (x+b/2)^2-b^2/4$$ holds over any ring in which $2$ has a multiplicative inverse. This means that we can reduce the problem of solving $x^2+bx+c = 0$ to the problem of finding the square roots of $b^2/4-c$.

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  • $\begingroup$ The ring isomorphism given in that WP-page reduces the problem of finding $n$th roots of unity in this ring to the same problem in the componentwise ring $\Bbb{R}^2$. In other words, there aren't any more roots of unity. The same approach can be applied to other polynomial equations. They are reduced to a system of two polynomial equations with real coefficients and unknowns. $\endgroup$
    – Jyrki Lahtonen
    Aug 12, 2018 at 6:45

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