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Let $V$ be a vector space over the field $\mathbb{Z}/p\mathbb{Z}$ with $p$ (obviously) prime. Is it possible to define an inner product and/or norm over $V$?

I am curious if this is possible. I am getting a bit confused about the positive-definiteness axioms of inner products. Every member of $\mathbb{Z}/p\mathbb{Z}$ is greater than or equal to $0$, but does this cause trouble? If this is not possible, why?

Ty Jensen
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    The field $\mathbb{Z}/p\mathbb{Z}$ (it is standard to put the $p$ before the second occurrence of $\mathbb{Z}$, not after) is not an ordered field. – Geoffrey Trang Jun 30 '20 at 01:15
  • @GeoffreyTrang could you explain why a field that is not ordered cannot have an inner product? – Ty Jensen Jun 30 '20 at 01:17
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    Because the definition of "inner product" requires there to be a notion of "nonnegativity" in the base field. – Geoffrey Trang Jun 30 '20 at 01:33
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    Sometimes "inner product" is used to mean just a nondegenerate symmetric bilinear form, and you can study these over finite fields. See other posts like this. – blargoner Jun 30 '20 at 01:47

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