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.

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
$$
{1}to
mathbb{Z}/nmathbb{Z}(1)to
{bar F}^timesto
{bar F}^timesto
{1}
$$
of discrete $Gamma$-modules, where $mathbb{Z}/nmathbb{Z}(1)$ 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 $delta_1:F^times/F^{times n}to H^1(Gamma,mathbb{Z}/nmathbb{Z}(1))$. Cup product on cohomology
$$
smile;:H^r(Gamma,mathbb{Z}/nmathbb{Z}(r))
times H^s(Gamma,mathbb{Z}/nmathbb{Z}(s))to
H^{r+s}(Gamma,mathbb{Z}/nmathbb{Z}(r+s))
$$
then provides a bilinear map
$
delta_2:F^times/F^{times n}times F^times/F^{times n}to
H^2(Gamma,mathbb{Z}/nmathbb{Z}(2)).
$

Lemma (Tate, 1970)
The map $delta_2(x,y)=delta_1(x)smiledelta_1(y)$ is a
symbol on $F$.

A symbol is a bilinear map $s:F^timestimes F^timesto A$ to a commutative
group such that $s(x,y)=0$ whenever $x+y=1$ in $F^times$.
There is a universal symbol $F^timestimes F^timesto K_2(F)$, giving rise
to Milnor's theory of higher $K$-groups $K_r(F)$ for every $rinmathbb{N}$,
as explained in Milnor's book.

This symbol also gives rise to a homomorphism
$$
delta_r:K_r(F)/nK_r(F)to
H^r(Gamma,mathbb{Z}/nmathbb{Z}(r)).
$$

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 $rinmathbb{N}$.

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 $pneq0$ 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 $oplus_r H^r(Gamma,mathbb{Z}/nmathbb{Z}(r))$ is generated by
elements of degree 1. Galois groups should thus be very special among
profinite groups in this respect.