deletions | additions       diff --git a/Description/K-theorydescription.tex b/Description/K-theorydescription.tex  index 357c86c..2153a39 100644  --- a/Description/K-theorydescription.tex  +++ b/Description/K-theorydescription.tex 
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If $F$ is an algebraically closed field, then for $n\geq1$, $K_n(F)$ is divisible and the torsion subgroup in $K_n(F)=0$ if $n$ is even and isomorphic to $\coprod_{l\neq char F}\mathbb{Q}_l/Z_{l}(n)$ if $n$ is odd.   \end{theo}   For The key ingridienst to understand this proof are  the case when $F$ is the algebraic closure work  of a finite field. We sill study Quillen around  the $K$ theory K-theory  for finite fields following thework of Quillen.  Also it will be interesting has a discusion around the Borel's theorem on the calculation of $K_{*}(F)\times\mathbb{Q}$ for number fields. Quillen shows that for a number field $K_n(F)$ is finitely generated, the ranks of this groups has interesting arithmetic information.  \end{document}