Quaternions and Fields

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Do the set of quaternions with appropriate operation obey all the laws of a field?

-- Joshua Taylor (The_jokeless_jester@hotmail.com), August 11, 2004



In math, a field is a set of numbers equiped with two operations, addition and multiplication. A field is a group under addition with zero as the identity operator. Quaternions with zero omitted are also a group under multiplication. R, C, and H are all division algebras. All three are associative fields. The real numbers are an ordered field, but neither complex numbers or quaternions are ordered. Both real and complex numbers are commute, but not so with quaternions. Basically, as the dimensions go up, properties of the real number field get lost. Here is a summary table:

Field Div. Alg Assoc. Commutes Ordered R + + + + C + + + - H + + - - Octonions + - - -

The big thing missing is a useful definition of the quaternion derivative of a quaternion-valued function. You can look at my suggestion on quaternions.com if that sort of thing is of interest.

-- Doug Sweetser (sweetser@alum.mit.edu), September 09, 2004.

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