228 CHAPTER 20
course, on the particular rotation.) This gives a linear action of A
:Foranyσ in A
, r(σ )actsonR
by rotating it.
We have seen that G, the absolute Galois group of Q,hasa
permutation representation on the roots of any given Z-polynomial.
Similarly, G has a permutation representation on the Q
any Z-variety W, deﬁned by permuting all of the solutions in Q
the system of equations deﬁning W. This works ﬁne when studying
Z-varieties deﬁned by a single polynomial in a single variable.
But when the varieties get more complicated, these permutation
representations are very hard to work with. We want to relate them
to some linear representations of G to get a better handle on the
number-theoretic properties of the variety W.
What is needed is a way to replace the permutation representa-
tion of G on the Q
points of W by a related linear representation
of G on some set of vectors associated with W.Thisiswhat
cohomology does. The process of replacing a complicated object by a
simpler linear object is called linearization.Itiswhatmustbedone
to Z-varieties in order to get the Galois representations we want.
For a much simpler example of this process of linearization,
consider the old problem of ﬁguring out your momentary speed
while driving along a highway. If you are driving at a constant speed
c, and you graph your distance driven versus the time, you will get
a straight line with slope c. So the slope is your speed. But if you are
speeding up and slowing down, the graph will be a curve. What is
your speed exactly now (say at t = 10.2 seconds after you started)?
To answer this question you have to replace the curved graph
by the straight line that approximates it most closely just at time
t = 10.2. This is the tangent line to the graph at that point. It has a
slope, and that slope is your instantaneous speed at time 10.2.
We say that we have linearized the graph of distance versus
time at that point by replacing it with the tangent line. If you are
interested only in your motion in a very small time frame around
t = 10.2, then this line will give a good approximation. This idea can