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Given an arbitrary vector space $V$ with a basis over an arbitary field $F$ is it possible to produce a norm on V? I know that if $V$ is a vector space with a basis over a field $K\subset \mathbb{C},\mathbb{R}$ then one can use $||\sum \alpha_i x_i||=\sum|\alpha_i|$ if the basis is finite and $||\sum \alpha_i x_i||=\max{\alpha_i}$ if the basis is infinite, but what about a field $F\not\subset \mathbb{C},\mathbb{R}$ ?

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Generally it is impossible. Consider a field $F$ whose characteristic $\ne 0$, you can add up identical non-zero vectors to get a zero vector. This violates the subadditivity of norm.

Edit: I don't think the absolute homogeneity of norm is even well-defined (or definable) if $F$ is arbitrary, so...

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The answer of @BigbearZzz is wrong. Actually, always it is possible to define a norm in a vector space even when the characteristic of the field is nonzero. For more details, see my answer to a similar question here: https://math.stackexchange.com/a/2568231/113061

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