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

  1. Jul 17, 2012 #1
    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...

    [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?
  2. jcsd
  3. Jul 17, 2012 #2

    ...and then [itex]\,G'\,[/itex] is not only "a set": it must be both an abelian group and a module over both fields [itex]\,F\,,\,G\,[/itex] ...

    If the module structures over both fields (module over field = vector space, of course) are preserved under the isomorphism

    (of rings) [itex]\,F\cong G\,[/itex], then yes: [itex]\,G'_F\cong G'_G[/itex] .

    All the algebraic invariants, under the above assumptions I wrote, are preserved: linearly independent

    sets, dimensions, etc., and you're right about the definition of linear map but only if we insist in working with both vector

    spaces [itex]\,G'_F\,,\,G'_G\,[/itex] , something that seems to me superfluous and confusing.

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