Anisotropic torus

Algebraic affine group scheme 24 Algebraic matrix group 29 Anisotropic torus 56 Anti-equivalence 15 Antipode 8 Arf invariant 147 Artin-Schreier theory 143 Augmentation ideal 13 Automorphism group scheme 58... 

We can use the previous theorem to show that every torus is nearly made up of two extreme types. Call a torus split (deploye) if it is actually diagonalizable, or in other words the Galois action on the character group is trivial. At the other extreme, call it anisotropic if it has no nontrivial maps to Gm, or in other words zero is the only fixed element in the character group. 

Theorem. Every torus T has a largest split subtorus Td and a largest anisotropic subtorus Ta. The intersection Td n Ta is finite, and T equals Ta - Td in the sense that no proper closed subgroup contains them both. 

Show that over the reals a torus T is anisotropic iff it is a product of copies of the circle group. [Let a Z" - Z" be an automorphism of order 2 with no fixed elements diagonalize a over the rationals to show it is multiplication by —1.]... 

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