" tommy quaternion "

[tommy1729]

Tommy, to be honest I just think that given like that, it an seems arbitrary set of conditions.
I don't quite get the red line that is connecting all those properties together and connecting the various number systems nor what you are looking for. I believe this is part of some quest you are undertaking, but I believe you have not made the goal explicit. What you are going for Tommy?

Sure it's a field in its own interesting and it "deserves more attention" but in giving these definition your going random or following some kind of map/logic?

I ask because there are infinite kinds of algebraic structures, and since it is abstract algebra, if you have not a chart is easy to get lost into meaningless abstraction (category theory was born for this reason, in order to do not get lost into Hilbert/Bourbaki's kind of structural abstractness, in that it gives you conceptual tool for mapping and exploring the territory).

Im still researching it.
I did not want to flood this thread with mini ideas and mini results.

My apologies for being vague , but I want to define things formally without being inconsistant.

The basic ideas are

1) unital and commutative but nonassociative numbers.
2) power-associative numbers so we can use taylor theorems.
3) no nilpotent elements
4) every element has at least 1 square root.
5) the smallest ones
6) no subnumbers only real coefficients. and not iso to an extension of 2 type of numbers ( like complex coefficients or other extensions of smaller dimensions )

then there are 2 cases left

the units sum to 0.

the units are linear independant.

assuming solutions exist ofcourse.  I conjecture yes.

On the other hand I conjecture only a finite amount of them ... probably between 0 and 3.
And all solutions having dimension below 28.

The 8 dimensional number given here has nilpotent elements. So it violates one of the conditions.
They always have a square root though.

I will post a candidate soon.

I was not able to find this relatively simple idea in the books.

I see applications in physics and math as I believe they are the " next quaternion ".

regards

tommy1729