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Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

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What fields can be used for an inner product space?

I do not know if any other fields are possible. Of course, an ordered field may or may not be a subfield of $\mathbb{R}$. … This immediately excludes finite fields. The basefield has to have additional structure, such as a distinguished automorphism. …