Editing Nimber (section)

== Multiplication ==
Nimber multiplication ('''nim-multiplication''') is defined recursively by

<math display=block> \alpha \, \beta = \operatorname{mex} \! \bigl(\{\alpha' \beta \oplus \alpha \, \beta' \oplus \alpha' \beta' : \alpha' < \alpha, \beta' < \beta \} \bigr).</math>

Nimber multiplication is associative and commutative, with the ordinal {{math|1}} as the multiplicative [[identity element]]. Moreover, nimber multiplication [[Distributive property|distributes over]] nimber addition.<ref name=Unity2021/>

Thus, except for the fact that nimbers form a [[class (set theory)|proper class]] and not a set, the class of nimbers forms a [[ring (algebra)|ring]]. In fact, it even determines an [[algebraically closed field]] of [[characteristic (algebra)|characteristic]] 2, with the nimber multiplicative inverse of a nonzero ordinal {{mvar|α}} given by

<math display=block>\alpha^{-1}  = \operatorname{mex}(S),</math>
where {{mvar|S}} is the smallest set of ordinals (nimbers) such that
# {{math|0}} is an element of {{mvar|S}};
# if {{math|0 < ''α''′ < ''α''}} and {{mvar|β'}} is an element of {{mvar|S}}, then <math>\tfrac{1 + (\alpha' \oplus \alpha) \beta'}{\alpha'}</math> is also an element of {{mvar|S}}.

For all natural numbers {{mvar|n}}, the set of nimbers less than {{math|2<sup>2<sup>''n''</sup></sup>}} form the [[Galois field]] {{math|GF(2<sup>2<sup>''n''</sup></sup>)}} of order&nbsp;{{math|2<sup>2<sup>''n''</sup></sup>}}. Therefore, the set of finite nimbers is isomorphic to the [[direct limit]] as {{math|''n'' → ∞}} of the fields {{math|GF(2<sup>2<sup>''n''</sup></sup>)}}. This subfield is not algebraically closed, since no field {{math|GF(2<sup>''k''</sup>)}} with {{mvar|k}} not a power of 2 is contained in any of those fields, and therefore not in their direct limit; for instance the polynomial {{math|''x''<sup>3</sup> + ''x'' + 1}}, which has a root in {{math|GF(2<sup>3</sup>)}}, does not have a root in  the set of finite nimbers.

Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that

# The nimber product of a Fermat 2-power (numbers of the form {{math|2<sup>2<sup>''n''</sup></sup>}}) with a smaller number is equal to their ordinary product;
# The nimber square of a Fermat 2-power {{mvar|x}} is equal to {{math|3''x''/2}} as evaluated under the ordinary multiplication of natural numbers.

The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal {{mvar|ω<sup>ω<sup>ω</sup></sup>}}, where {{mvar|ω}} is the smallest infinite ordinal. It follows that as a nimber, {{mvar|ω<sup>ω<sup>ω</sup></sup>}} is [[transcendental number|transcendental]] over the field.<ref>Conway 1976, p.&nbsp;61.</ref>