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Fearless Symmetry by Robert Gross, Avner Ash

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THE RESTRICTION MORPHISM 161
same thing to every root of every Z-polynomial. So γ and γ
are the
same.
The importance of the restriction morphisms cannot be over-
stressed. The absolute Galois group is infinite, and we humans
cannot grasp it all at once. All we see are its manifestations in our
human world, namely, how it acts, via the restriction morphisms,
on the various Galois groups of single Z-polynomials. At least
theoretically, we can understand any single Z-polynomial if we put
our minds to it. We cannot grasp G all at once. But by seeing what it
does to the roots of individual Z-polynomials, we can hope to prove
theorems about G.
In fact, we can sometimes do more. Sometimes there are whole
infinite families of Z-polynomials we can study together. For
example, there is the family x
n
1, where n = 1, 2, 3, .... We will
look more closely at these in chapter 18 when we investigate one-
dimensional Galois representations.
Examples
We finish this chapter with a couple of examples. The only elements
in G that we can describe totally explicitly are the neutral element
ι and complex conjugation c. So we will give one example for each.
EXAMPLE: Whatever Z-polynomial f may be, r
G( f )
(ι)isalways
the identity permutation.
EXAMPLE: Refer to the example in the last chapter where
f (x) = x
3
5 and use the notation introduced there for its
three roots, r, s, t: r =
3
5, s = ω
3
5, and t = ω
2
3
5. Now,
c(
3
5) =
3
5because
3
5 is real. Also, c(ω) = ω
2
,andc(ω
2
) = ω
(which you can deduce from the fact that ω and ω
2
are the two
roots of x
2
+ x + 1, or you can work it out directly from the
value of ω given in the previous chapter).
So you see that r
G( f )
(c) performs the permutation r r,
s t,andt s.

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