Editing Cayley–Dickson construction (section)

==Synopsis==

{| class="wikitable floatleft"
|+ Cayley–Dickson algebras properties
|-
! rowspan=2|[[Algebra over a field|Algebra]]
! rowspan=2|[[Dimension (vector space)|Dimension]] !! rowspan=2|[[Ordered field|Ordered]] !! colspan=4|[[Multiplication]] properties !! rowspan=2|Nontriv.<br>[[zero divisor|zero<br>divisors]]
|-
! [[Commutative]] !! [[Associative]] !! [[Alternative algebra|Alternative]] !! [[Power associativity|Power-assoc.]]
|-
! [[Real number|Real&nbsp;numbers]]
| style="text-align:center" | 1 || {{Yes}} || {{Yes}} || {{Yes}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Complex number|Complex&nbsp;num.]]
| style="text-align:center" | 2 || {{No}} || {{Yes}} || {{Yes}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Quaternion]]s
| style="text-align:center" | 4 || {{No}} || {{No}} || {{Yes}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Octonion]]s
| style="text-align:center" | 8 || {{No}} || {{No}} || {{No}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Sedenion]]s
| style="text-align:center" | 16 || {{No}} || {{No}} || {{No}} || {{No}} || {{Yes}} || {{Yes}}
|-
! [[Trigintaduonion]]s<br>and higher
| style="text-align:center" | ≥ 32 || {{No}} || {{No}} || {{No}} || {{No}} || {{Yes}} || {{Yes}}
|}{{clear}}

The '''Cayley–Dickson construction''' is due to [[Leonard Dickson]] in 1919 showing how the [[octonion]]s can be constructed as a two-dimensional algebra over [[quaternion]]s. In fact, starting with a field ''F'', the construction yields a sequence of ''F''-algebras of dimension 2<sup>''n''</sup>. For ''n'' = 2 it is an associative algebra called a [[quaternion algebra]], and for ''n'' = 3 it is an [[alternative algebra]] called an [[octonion algebra]].  These instances ''n'' = 1, 2 and 3 produce [[composition algebra]]s as shown below. 

The case ''n'' = 1  starts with elements (''a'', ''b'') in ''F'' × ''F'' and defines the conjugate (''a'', ''b'')* to be (''a''*, –''b'') where ''a''* = ''a'' in case ''n'' = 1, and subsequently determined by the formula. The essence of the ''F''-algebra lies in the definition of  the product of two  elements (''a'', ''b'') and (''c'', ''d''):
:<math>(a,b) \times (c,d) = (ac - d^*b, da + bc^*).</math>
'''Proposition 1:''' For <math>z = (a,b)</math> and <math>w = (c,d),</math> the conjugate of the product is <math>w^*z^* = (zw)^*.</math>
:proof: <math>(c^*,-d)(a^*,-b) = (c^*a^* + b^*(-d), -bc^*-da) = (zw)^*.</math>
'''Proposition 2:''' If the ''F''-algebra is associative and <math>N(z) = zz^*</math>,then <math>N(zw) = N(z)N(w).</math>
:proof: <math>N(zw) = (ac-d^*b, da+bc^*)(c^*a^*-b^*d, -da -bc^*) = (aa^* + bb^*)(cc^* + dd^*)</math> + terms that cancel by the associative property.