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\section{Plan of the Talks}  \subsection{Definition \subsection{Introduction to K-theory}  \paragraph{Definition  of $K_0$, $K_1$ and of higher $K_i$ via Quillen's plus construction} construction.}  In this talk we will review $K_0$, $K_1$ and $K_2$ of rings.  \begin{itemize} 
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\item Remark $K$-theory with finite coeficients.  \end{itemize}  \subsection{Norm-Residue Homomorphism} Homomorphism (Merkurjev-Suslin theorem)}  \paragraph{Construction of the norm-residue homomorphism}  \subsection{Borel's theorem}  \subsection{Rank \paragraph{Rank  of $K_n$ over number fields} \begin{itemize}  \item Show how to reduce the problem to the computation of the real cohomology of $SL(F)$. Borel determined the ring $H^*(SL_m(R);\mathbb{Q})$ and its dual coalgebra. Relate the primitive elementes of to the idecomposables elementes and finally to $H^*(SL(O_F),\mathbb{R})$. 
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The pourpuse of this talk is assuming the computation of the real cohomology of $SL(F)$ prove Theorem \ref{Borel-Ranks}.  \end{itemize}  \subsection{The \paragraph{The  Borel regulator} regulator.}  (Burgos Gil Book)  \subsection{$K_3$ of fields and the relation with the Bloch group} \subsection{Rigidity theorem}  \subsection{K-theory \paragraph{K-theory  for finite fields} Quillen's work in Adam's conjecture  \subsection{K-theory \paragraph{K-theory  of algebraically closed fields} The pourpose purpose  of this section is prove one of the Quillen-Lichtembaum 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 evan and isomorphic to $\coprod_{l\neq char F}\mathbb{Q}_l/Z_{l}(n)$ if $n$ is odd.  
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(Second Talk of this part) From the preliminar results describe the structure of $K_*(F)$   Corollary 1.3.1 and Proposition 1.4   The main references for this part are: Weibel's book \cite{Kbook} VI.1 and Suslin's paper  \paragraph{$K_3$ of fields and the relation with the Bloch group}  \bibliographystyle{alpha}