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  "body": "\n## Summary\n\nAn algebraic torus is a [[linear algebraic group|GA81VVQHEP]] that becomes a power of the multiplicative group $\\mathbb{G}_m$ over an algebraic closure of the base field.\nOver an algebraically closed field it is a group of invertible diagonal matrices.\nMaximal tori control the structure of [[reductive groups|2XS84PWE3S]] through their characters and roots.\nTori also give [[toric varieties|YWB981MEF2]].\n\n## Explanations\n\n### L1 human\n\nThe simplest torus is the set of nonzero numbers under multiplication.\nA torus of dimension $n$ is the set of $n$-tuples of nonzero numbers, multiplied coordinatewise, or equivalently the invertible diagonal $n \\times n$ matrices.\nOver the real numbers there are also twisted tori, such as the circle $x^2 + y^2 = 1$ with the multiplication of complex numbers $x + iy$.\nOver $\\C$ the circle becomes the same as $\\C^{*}$ through $(x, y) \\mapsto x + iy$.\n",
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