View source for Algebraic torus

An [[Algebraic group|algebraic group]] that is isomorphic over some extension of the ground field to the direct product of a finite number of multiplicative groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117101.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117102.png" /> of all algebraic homomorphisms of an algebraic torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117103.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117104.png" /> is known as the character group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117105.png" />; it is a free Abelian group of a rank equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117106.png" />. If the algebraic torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117107.png" /> is defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117108.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a0117109.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171010.png" />-module structure, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171011.png" /> is the Galois group of the separable closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171012.png" />. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171013.png" /> defines a duality between the category of algebraic tori over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171014.png" /> and the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171015.png" />-free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171016.png" />-modules of finite rank. An algebraic torus over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171017.png" /> that is isomorphic to a product of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171018.png" /> over its ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171019.png" /> is called split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171020.png" />; any algebraic torus over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171021.png" /> splits over a finite separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011710/a01171022.png" />. The role played by algebraic tori in the theory of algebraic groups greatly resembles the role played by tori in the theory of Lie groups. The study of algebraic tori defined over algebraic number fields and other fields, such as finite fields, occupies an important place in problems of arithmetic and in the classification of algebraic groups. Cf. [[Linear algebraic group|Linear algebraic group]]; [[Tamagawa number|Tamagawa number]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Ono,   "Arithmetic of algebraic tori"  ''Ann. of Math. (2)'' , '''74''' :  1  (1961)  pp. 101–139</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Ono,   "On the Tamagawa number of algebraic tori"  ''Ann. of Math. (2)'' , '''78''' :  1  (1963)  pp. 47–73</TD></TR></table>