## Related pagesMoufang planesInverse property Lenz type VI |

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

**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).

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