Octonion algebra

<h2 id="classification">Classification</h2>
It is a theorem of <a href="/facts/Adolf_Hurwitz/urNgHh58">Adolf Hurwitz</a> that the F-<a href="/facts/Isomorphism_class/wDUgsJQ8">isomorphism classes</a> of the norm form are in one-to-one correspondence with the isomorphism classes of octonion F-algebras. Moreover, the possible norm forms are exactly the <a href="/facts/Pfister_form/wsNb9mFC">Pfister 3-forms</a> over F.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
Since any two octonion F-algebras become isomorphic over the <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of F, one can apply the ideas of non-<a href="/facts/Abelian_group/FfWwmx4R">abelian</a> <a href="/facts/Galois_cohomology/nzC1EVYT">Galois cohomology</a>. In particular, by using the fact that the <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> of the split octonions is the split <a href="/facts/Algebraic_group/DDtBCIvh">algebraic group</a> <a href="/facts/G2_(mathematics)/Bka4pKjQ">G2</a>, one sees the correspondence of isomorphism classes of octonion F-algebras with isomorphism classes of G2-<a href="/facts/Torsor/N27maCdL">torsors</a> over F. These isomorphism classes form the non-abelian Galois cohomology set 
 
 
 
 
 H
 
 1
 
 
 (
 F
 ,
 
 G
 
 2
 
 
 )
 
 
 {\displaystyle H^{1}(F,G_{2})}
 
.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<ul><li><a href="/facts/Skip_Garibaldi/8rCvXUs9">Garibaldi, Skip</a>; <a href="/facts/Alexander_Merkurjev/DowcduN3">Merkurjev, Alexander</a>; <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, Jean-Pierre</a> (2003). Cohomological invariants in Galois cohomology. University Lecture Series. Vol. 28. Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-3287-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1159.12311">1159.12311</a>.</li>
<li><a href="/facts/Tsit_Yuen_Lam/S1ewInkE">Lam, Tsit-Yuen</a> (2005). Introduction to Quadratic Forms over Fields. <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>. Vol. 67. <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1095-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2104929">2104929</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1068.11023">1068.11023</a>.</li>
<li>Okubo, Susumu (1995). Introduction to octonion and other non-associative algebras in physics. Montroll Memorial Lecture Series in Mathematical Physics. Vol. 2. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. p. 22. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-47215-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0841.17001">0841.17001</a>.</li>
<li>Schafer, Richard D. (1995) [1966]. <a href="https://archive.org/details/introductiontono0000scha">An introduction to non-associative algebras</a>. <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-68813-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0145.25601">0145.25601</a>.</li>
<li>Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) [1978]. Rings that are nearly associative. <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-779850-1. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0518614">0518614</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0487.17001">0487.17001</a>.</li>
<li><a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Serre, J. P.</a> (2002). Galois Cohomology. Springer Monographs in Mathematics. Translated from the French by Patrick Ion. Berlin: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-42192-0. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1004.12003">1004.12003</a>.</li>
<li><a href="/facts/T._A._Springer/xHUWxQYC">Springer, T. A.</a>; Veldkamp, F. D. (2000). Octonions, Jordan Algebras and Exceptional Groups. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-66337-1.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Cayley–Dickson_algebra">"Cayley–Dickson algebra"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Schafer (1995) p.48 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402, see 399 <a href="/wiki/Max_Zorn" target="_blank">/wiki/Max_Zorn</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Furey, C. (10 October 2018). "Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra". Physics Letters B. 785: 84–89. arXiv:1910.08395. Bibcode:2018PhLB..785...84F. doi:10.1016/j.physletb.2018.08.032. ISSN 0370-2693. <a href="https://doi.org/10.1016%2Fj.physletb.2018.08.032" target="_blank">https://doi.org/10.1016%2Fj.physletb.2018.08.032</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lam (2005) p.327 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Garibaldi, Merkurjev & Serre (2003) pp.9-10,44 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Octonion algebra open-in-new

Octonion algebra