I was working on a problem in field extensions (for a 3rd-year ring theory class), and came to a point where I essentially had the following situation...(adsbygoogle = window.adsbygoogle || []).push({});

[itex] F [/itex] is a field isomorphic to [itex] G [/itex], and [itex] G' [/itex] is for all intents and purposes, some set. We can then consider the vector spaces [itex] G'_F [/itex] and [itex] G'_G [/itex].

I wondered to myself if it was true that [itex] G'_F [/itex] was isomorphic to [itex] G'_G [/itex]. However, in my mathematical career I've only ever worked with the notion of vector space isomorphisms over a fixed field.

I ended up solving the problem differently, but the question remained... What happens if you don't fix the field?

I suppose the first inherent problem with working with an unfixed field is that the definition of a linear map would have to be changed; it would have to be something like... for [itex] u,v\in V [/itex] and [itex]α[/itex] in the field, a linear map is something which satisfies [itex] L(αv+u)=l(α)L(v)+L(u) [/itex] where [itex] l [/itex] is a field homomorphism specified in the definition of [itex] L [/itex].

Aside from this however, has anybody ever tried doing this, and seeing if properties like dimension are preserved under a "suitable" definition of vector space isomorphism over an unfixed field?

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# Why do we only work with vector space isomorphisms over a fixed field?

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