Let $E/F$ be a finite separable extension of a commutative field $F$. Let $T$ be the torus ${\rm Res}_{E/F}\; {\mathbb G}_m$, where ${\rm Res}$ is Weil's restriction of scalar. Is there a simple description of all subtori of $T$ ?
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asked Dec 3, 2010 at 17:08
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1$\begingroup$ You can use the equivalence of categories between tori and free f.g. abelian groups plus Galois action to turn this in to a purely representation-theoretic question: "if $G$ is a finite group then what are the $G$-stable saturated (i.e. no torsion in cokernel) subgroups of $\mathbf{Z}[G]$". Then an algebraist might be able to help. $\endgroup$– Kevin BuzzardCommented Dec 3, 2010 at 19:48
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$\begingroup$ In fact, because they're saturated, aren't you just asking for $\mathbf{Q}[G]$-submodules of $\mathbf{Q}[G]$? And this is standard. $\endgroup$– Kevin BuzzardCommented Dec 3, 2010 at 19:49
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3$\begingroup$ Actually, because you didn't assume $E/F$ Galois the question is actually "what are the $\mathbf{Q}[G]$-submodules of $\mathbf{Q}[G/H]$? (the vector space with basis $G/H$ with $G$ acting on the left). This is still very well-understood. $\endgroup$– Kevin BuzzardCommented Dec 3, 2010 at 20:39
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