I can (barely) understand the definition of the higher algebraic K-groups a la the plus construction right now (I have some past familiarity with K-theory for C\*-algebras and can recall the rudiments of the situation for vector bundles). So by "simple", I mean to a mathematical layman. If you have a complicated answer, feel free to answer as well, but I probably won't be able to understand much.



Is there a simple relationship between K-theory and Galois theory?
Perhaps the other Bloch-Kato conjecture is more relevant; it relates *Milnor's* higher $K$-groups and Galois cohomology.


The following text is lifted from the expository account [on the arXiv.](http://arxiv.org/abs/math/0311099)


Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a
separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There
is an exact sequence
$$
\{1\}\to
\mathbb{Z}/n\mathbb{Z}(1)\to
{\bar F}^\times\to
{\bar F}^\times\to
\{1\}
$$
of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and
Hilbert's theorem 90 furnish an isomorphism $\delta\_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology
$$
\smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))
\times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to
H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s))
$$
then provides a bilinear map
$
\delta\_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to 
H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)).
$


**Lemma** (Tate, 1970)
*The map* $\delta\_2(x,y)=\delta\_1(x)\smile\delta\_1(y)$ *is a
symbol on* $F$.


A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative
group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$.
There is a universal symbol $F^\times\times F^\times\to K\_2(F)$, giving rise
to Milnor's theory of higher $K$-groups $K\_r(F)$ for every $r\in\mathbb{N}$,
as explained in Milnor's book.


This symbol also gives rise to a homomorphism 
$$
\delta\_r:K\_r(F)/nK\_r(F)\to
H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)).
$$


**Conjecture** (Bloch-Kato, 1986)
*The map*
$\delta\_r$ *is an isomorphism for all fields* $F$, *all integers* $n>0$
(*invertible in* $F$) *and all indices* $r\in\mathbb{N}$.


The main theorem of Merkurjev-Suslin (1982) says that the map
$\delta\_2$ is
always an isomorphism ; Tate had proved this earlier (1976) for global fields.
Bloch-Gabber-Kato prove this conjecture when $F$ is a field of
characteristic $0$ endowed with a henselian discrete valuation of residual
characteristic $p\neq0$ and $n$ is a power of $p$.


Somebody should ask a qustion about the current status of the Bloch-Kato
conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the
details is not available in one place. 


The Bloch-Kato conjecture makes the remarkable prediction that the graded
algebra $\oplus\_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by
elements of degree 1. Galois groups should thus be very special among
profinite groups in this respect.


