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Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dotsc, a_n) $ for a basis $\{1, i_1, \dotsc, i_n \}$ and we have a basis so that $ i_k^2 \in \{ -1, +1 \}$.

This implies the answer can be given in the form of a Cayley table for multiplying the finite dimensional basis $\{1, i_1, \dotsc, i_n \}$.

So our algebra is of the form $ a_0 + a_1 i_1 + a_2 i_2 + \dotsb$, addition is as usual and products are defined by a Cayley table relating the basis elements $\{1, i_1, \dotsc, i_n \}$ and we have a basis so that $ i_k^2 \in \{ -1, +1 \}$.

Also the product of two base elements $i_a$ and $i_b$ is never $0$.

Assume the algebra is also power-associative.

One important extra condition:

Let $T$ be such an Algebra.

Let $x$ be an element of $T$. Then $y^2 = x$ always has a solution $y$ within $T$.

So basically the square root is defined in the algebra.

The final condition is that this type of algebra $T$ we seek, is not isomorphic to a tensor product of the complex numbers with another algebra $B$ (taken over the reals).

What are simple or small examples of such "hypercomplex" algebras? What is the smallest dimension one can have?

I know there are methods* to test potential candidates (dimensions or proposed solutions) but they are cumbersome and give no “deeper insights”.

(*they usually involve things like Jacobi matrix or hessian)

I can’t find examples in literature. So I’m not even sure they exist but I hope they do.

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    $\begingroup$ Your first and third paragraphs seem to say substantially the same thing. \\ Also, the condition that every element has a square root is different from what I would expect when you say "the square root is defined in the algebra"; I would take the latter to mean that there is a distinguished function $\sqrt\cdot : T \to T$ (as your title suggests) satisfying the obvious equation, and possibly some additional conditions. Depending on those additional conditions, this is tronger than what you wrote. $\endgroup$ – LSpice Mar 22 at 0:58
  • $\begingroup$ @LSpice : Im a bit confused by your comments. Are you suggesting my question has different interpretations possible ? Or has contradiction or unclear parts ? What is the distinction you are trying to make ? Notice the dimension is finite. $\endgroup$ – mick Mar 22 at 23:41
  • $\begingroup$ I'm saying that, for example, every element of $\mathbb C$ has a square root, but there is no reasonable sense of a square-root function $\sqrt\cdot : \mathbb C \to \mathbb C$ without making arbitrary and incoherent choices; so I was wondering whether you really just wanted the existence of square roots elementwise, or you wanted a square-root function satisfying certain coherence conditions. \\ As to "Your first and third paragraphs seem to say substantially the same thing", I mean exactly that. I think you can delete the third paragraph with no change in meaning. $\endgroup$ – LSpice Mar 23 at 4:33
  • $\begingroup$ @LSpice ah ok. Yeah I agree. I just want the existence of the square root elementwise. And yeah I kinda repeated myself in the third paragraph. It was intented for clarity. $\endgroup$ – mick Mar 23 at 12:01

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