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 $T$ ?

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.
–
Kevin BuzzardDec 3 '10 at 19:48

In fact, because they're saturated, aren't you just asking for $\mathbf{Q}[G]$-submodules of $\mathbf{Q}[G]$? And this is standard.
–
Kevin BuzzardDec 3 '10 at 19:49

3

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.
–
Kevin BuzzardDec 3 '10 at 20:39

Thanks a lot Kevin, this helps me !
–
Paul BroussousDec 4 '10 at 11:59