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%\date{} % Activate to display a given date or no date  \begin{document}  The localization Theorem and the Gersten conjecture ofen reduce general problems of Algebraic-K theory to the particular case of fields. The  aim of the seminar will be understand the $K$-theory of fields, we will recall some of the constructions of the higher $K$-groups and  achieved by Quillen andsome of  its basic properties as Localization theorem, Gersten's conjecture Projective Bundle Formula;  but instead in focus in the theoretical proofs we will study explicit computations around the $K$-theory fields. One of the aims is study the proof of uone one  of the Qullien-Lichtenbaum conjectures: \begin{theo}  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}   The key ingridienst to understand this proof are which follows from  the work first version  of Quillen the Suslin's Rigidity Theorem and from the Quillen's work  around the computation of then  K-theory for of  finite fields following the fields.  Also it will be interesting has a discusion around the Borel's theorem on the calculation of $K_{*}(F)\otimes\mathbb{Q}$ 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.