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Showing that a field is isomorphic to a field of vectors

  1. Apr 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Let [itex]K / F[/itex] be a field extension of degree [itex]n[/itex].

    For any [itex]a \in K[/itex] prove that the map defined by [itex]\sigma_a(x) = ax[/itex] is a linear map over the vector space [itex]K / F[/itex]. This part I understand.

    Show that [itex]K[/itex] is isomorphic to the subring [itex]F^{n x n}[/itex] of [itex]n x n[/itex] matrices with entries in [itex]F[/itex].

    3. The attempt at a solution
    I understand that linear map from a field extension to itself = an [itex]n x n[/itex] matrix. The trick is showing that the operations of addition and multiplication are compatible, and I can't seem to do that without a lot of tedious, convoluted calculations, especially since there's not a good closed form for the transformation matrix that would apply to all bases (as far as I know). Is there a better way?
     
  2. jcsd
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