<sod>Later work showed that in fact, any finite-dimensional real division algebra must be of dimension 1, 2, 4, or 8. This was independently proved by [[Michel Kervaire]] and [[John Milnor]] in 1958, again using techniques of [[algebraic topology]], in particular [[K-theory]]. [[Adolf Hurwitz]] had shown in 1898 that the identity <math>q\overline{q} = \textrm{sum\ of\ squares}</math> held only for dimensions 1, 2, 4 and 8.<ref>{{cite book|title=[[The Road To Reality]]|authorlink=Roger Penrose|author=Roger Penrose|year=2005|publisher=Vintage|isbn=0-09-944068-7}}</ref> (See [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]].)<eod>
<sod>Later work showed that in fact, any finite-dimensional real division algebra must be of dimension 1, 2, 4, or 8. This was independently proved by [[Michel Kervaire]] and [[John Milnor]] in 1958, again using techniques of [[algebraic topology]], in particular [[K-theory]]. [[Adolf Hurwitz]] had shown in 1898 that the identity <math>q\overline{q} = \textrm{sum\ of\ squares}</math> held only for dimensions 1, 2, 4 and 8.<ref>{{cite book|title=[[The Road To Reality]]|authorlink=Roger Penrose|author=Roger Penrose|year=2005|publisher=Vintage|isbn=0-09-944068-7}}, p.202</ref> (See [[Hurwitz's theorem (normed division algebras)|Hurwitz's theorem]].)<eod>