Wikipedia says we can't have inner products on vector spaces over finite fields. Can we have an inner product on a vector space over $\mathbb Q$? What about an inner product on vector spaces over intermediate fields between $\mathbb Q$ and $\mathbb C$?
(Questions like this or this don't make it clear to me if it's possible).
Note that the ordinary inner product $\langle \cdot,\cdot\rangle: \mathbb C^n \times \mathbb C^n \to \mathbb C$ restricts nicely: for each subfield $K \subseteq \mathbb C$ which is closed under complex conjugation, we have that $\langle \cdot ,\cdot\rangle|_{K^n \times K^n}: K^n \times K^n \to K$, and it still trivially satisfies all the requirements of an inner product.