MilKThy.tex

\begin{definition}{Milnor $\K$-theory}{}
	For any field $\FF$, define $\forall x\in\FF^{\times}$,
	\[\K_\bullet^\Mil(\FF)=T(\FF^\times)/x\otimes(1-x)\]
	to be the Milnor $\K$-theory of $\FF$, which is a graded algebra, where $T(X)$ is 
\end{definition}
For example, $\K_0^\Mil(\FF) = \ZZ$, $\K_1^\Mil(\FF) = \FF^\times$. 

\begin{proposition}{}{}
	\begin{itemize}
		\item $[x][y]+[y][x]=0$;
		\item $[x][x]=[x][-1]$. 
	\end{itemize}  
\end{proposition}

\begin{proof}
1) $[x][-x]=[x]\left[\frac{1-x}{1-x^{-1}}\right]$
\[
\begin{aligned}
& =[x][1-x]+\left[x^{-1}\right]\left[1-x^{-1}\right] \\
& =0
\end{aligned}
\]
\[
\begin{aligned}
S_0[x][y]+[y][x]= & {[x][-x]+[x][y]+[y][x] } \\
& +[y][-y] \\
= & {[x](-x y]+[y][-x y] } \\
= & (x y][-x y]=0
\end{aligned}
\]
2) $[x][x]=[x][-1]+[x][-x]=[x][-1] . D$
\end{proof}
\begin{proposition}
	Let $\Bbbk$ be a field and $V$ be a normalized discrete valuation on $\Bbbk$. Let $\Bbbk(V)=\OoO_V/m_V$ be
\end{proposition}
Prop 3.3 Let $k$ be a field and $v$ be a normalized discrete valuation on $k$. Let $R(v)=0 \mathrm{r} / \mathrm{m}_v$ be the residue field 1 . Then $\exists$ ! homomorphism
Sit. $\forall u_1, \ldots, u_{n-1} \in k_n^m(k)-k_{n-1}^m(k(v))$ $\partial_v\left([x]\left[u_1\right]\right.$ and $x \in K^x$
whee $\bar{u}_i$ is the (lass of $u_i$ in $R(v)^x$.
Prof. The uniqueness is clear. For exist. pence, choosing a uniformizer $\pi$, define a graded ring orphism