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In the 'Field' article at MathWorld there is a phrase that I don't understand:

It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements are equivalent to the field of complex numbers.

The same sentence occurs here, at the description of the 'field-theory' tag.

What does this exactly mean? What is the field concept applied to triplets of elements? Which theorem this phrase refers to?

John McClane
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  • Someone should edit MathWorld. If you try to define a multiplication on the set of all ordered pairs of real numbers in such a way that, together with addition defined component-wise, you get a field, then you get (a structure isomorphic to) the complex numbers. Hamilton tried to do this with ordered triples of real numbers, with no success, but found he could do it with ordered quadruples, if he was willing to abandon commutativity of the multiplication. See also https://www.quora.com/Why-do-we-have-analysis-of-ordered-pairs-complex-analysis-but-not-ordered-triples – Gerry Myerson Oct 17 '18 at 12:18
  • Any thoughts, John? – Gerry Myerson Oct 20 '18 at 23:10
  • @Gerry Myerson I think we should burninate this sentence from the description of the 'field-theory' tag. At least, the word triplets looks suspicious there. Thank you for the link to the Quora question, it clarified things to some extent. Unfortunately, not a word about Hilbert or Weierstrass, so the question remains open. – John McClane Oct 21 '18 at 14:04
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    Some more links of possible interest (but no mention of Hilbert/Weierstrass): https://math.stackexchange.com/questions/1784166/why-are-there-no-triernions-3-dimensional-analogue-of-complex-numbers-quate and https://math.stackexchange.com/questions/865130/proving-that-mathbb-r3-cannot-be-made-into-a-real-division-algebra-and-that and https://math.stackexchange.com/questions/1547756/can-there-be-a-set-of-numbers-which-have-properties-like-those-of-quaternions? and https://math.stackexchange.com/questions/1610860/generalizing-complex-numbers-is-there-a-mathematical-system-isomorphic-to-3-dim? – Gerry Myerson Oct 21 '18 at 21:04
  • Also https://math.stackexchange.com/questions/529/why-are-the-only-division-algebras-over-the-real-numbers-the-real-numbers-the-c and https://math.stackexchange.com/questions/751106/complex-number-with-3-dimensions and probably many more. – Gerry Myerson Oct 21 '18 at 21:07

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