### Related pages

Moufang planes
Inverse property
Lenz type VI
An alternative field is a semifield K which satisfies the alternative laws

(ab)b=ab2 and a(ab)=a2b

for alle elements a,b in K. The alternative fields are exactly the ternary fields coordinatizing Moufang planes. In fact, the following theorem holds.

Theorem. A ternary field (K,T) is an alternative field if and only if the projective plane P(K,T) over K is a Moufang plane.

Given any alternative field K, the following algebraic facts are known.

• The multiplicative loop of K has the inverse property.
• K is biassociative, i.e. each two elements of K are contained in a subskewfield of K
• K is commutative if and only if K is a field.
• If K is finite, then K is a field.
• If K is not a skewfield, then
Moreover, the Bruck-Kleinfeld theorem describes all alternative fields.

Theorem. Let K be an alternative field. Then either K is a skewfield or K is an eight-dimensional octonion algebra over it's center Z(K).

Contributed by Hauke Klein
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