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Yesterday I was in a discussion about solving an applied problem using clifford algebras over a finite field. While this is not (seemingly) disallowed by the definition of a clifford algebra (which seems to either say it can be over a field or over a ring depending on who you consult), there is some ambiguity about how much sense an inner product makes on a finite field depending on the chosen finite field since it must induce a vector space.

This led to some ineffective googling where I tried to find examples of clifford algebras over finite fields: I could not!

So: Are there (applied?) examples of Clifford algebras over finite fields that are studied or are these generally uninteresting?

o1lo01ol1o
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    Clifford algebras are defined with quadratic forms, not inner products. In characteristic not 2, quadratic forms are equivalent to symmetric bilinear forms, of which an "inner product" is usually a particular type. There is no ambiguity here. I do not have any "natural" examples of such Clifford algebras, but constructing examples is easy. Probably you would want to look towards number theory. – Nicholas Todoroff Feb 28 '23 at 18:52
  • @NicholasTodoroff thanks for the clarification, the relation between inner product and symmetric bilinear form dispelled my sense of ambiguity. – o1lo01ol1o Mar 01 '23 at 09:17
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    If you scrutinize the definition of a quaternion algebra you'll see that they are 2D Clifford algebras, and I seem to be getting a lot of hits searching for quaternion algebras over finite fields. – Nicholas Todoroff Mar 01 '23 at 16:44

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