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Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ and $L/\mathbb{Q}$ a finite extension. Assume that $V$ is equipped with a structure of $L$-vector space. Then there is a decomposition $$ V \otimes_\mathbb{Q} \mathbb{C}=\bigoplus_{\sigma \in Hom(L, \mathbb{C})} V_\sigma $$

A) What is the best way to think of that?

I guess it has something to do with the fact that $L \otimes_\mathbb{Q} \mathbb{C}$ is isomorphic to $\mathbb{C}^{Hom(L, \mathbb{C})}$...

B) Is it true that all $V_\sigma$ have the same dimension? If so, how to prove it?

Thanks for your help.

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    $\begingroup$ A) $V \otimes_{\mathbb Q} L \otimes_L {\mathbb C}$ $\endgroup$
    – Allen Knutson
    Commented Jun 24, 2013 at 12:37
  • $\begingroup$ could you please elaborate a bit more? what about question B? $\endgroup$
    – compositio
    Commented Jun 24, 2013 at 13:50
  • $\begingroup$ When you write $V_\sigma$, do you mean $V \otimes_{L, \sigma} \mathbb{C}$? The latter notation makes the equidimensionality clear. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 24, 2013 at 14:01
  • $\begingroup$ If $L=\mathbb{Q}(\alpha)$ then $V_\sigma$ is the subspace of $V_\mathbb{C}$ where $\alpha v=\sigma(\alpha)v$ (this is independent from $\alpha$). Is that the same as $V \otimes_{L, \sigma} \mathbb{C}$? $\endgroup$
    – compositio
    Commented Jun 24, 2013 at 14:12

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