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A good treatment have been given to the quaternion equations. Indeed, Ivan Niven in his paper Equations in Quaternion given in this link gave an algorithm on how to find all the solution of a given quaternion polynomial equation. Actually, in later papers the results were further improved.

Now my question is, is there similar treatment or even research about the polynomial equations over the octonions. I have searched google but I reached nothing. So, I thought someone would be able to help me.

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You might have better luck searching for "octonions" rather than "octinions". – Robert Israel Apr 16 '12 at 22:28
I fixed the spelling. – Igor Rivin Apr 16 '12 at 23:13
up vote 2 down vote accepted

The paper

On Octonionic Polynomials Rogério Serôdio (in advances in applied clifford algebras -- who knew such a journal existed?) seems to address precisely the questions you ask.

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Thank you, the author assumes that the coefficients are real numbers only. I think I can get around that. I am trying, if you have any ideas or suggestions, please tell me – Motaz Hammouda Apr 17 '12 at 5:56

You can give a look at the paper

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect, by R. Ghiloni and A. Perotti

on Annali di Matematica Pura ed Applicata Volume 190, Number 3, 539-551, DOI: 10.1007/s10231-010-0162-1

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