Post #1815090
2026-04-07 18:49 UTC
@hallasurvivor How do you even define polynomials over quaternions? If we define it as a formal sum of aq^n where q is a variable, then evaluation is no longer a homomorphism. If we allow terms such as aqbqc, then factorization of a polynomial with root q0 into g(q) (q - q0) completely breaks, because division breaks (because like terms can't cancel, for example aqb - cqd doesn't simplify for generic quaternions a,b,c,d).
Replies (2)
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@trebor@types.pl 2026-04-07 18:57
@hallasurvivor Suppose we define the algebra of quaternionic polynomials H as formal sums of terms like aqbqcqd... where a,b,c,d are quaternions and q is the variable. Real numbers commute with everything, so this forms an R-algebra. Recall that evaluation of real polynomials R[x] -> R^R is an injective homomorphism. It is however not true for quaternions H -> H^H. What kind of identities should we add to H to make it injective?
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@hallasurvivor@sunny.garden 2026-04-07 23:00
@trebor this is exactly the issue! Nice ^_^