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Fearless Symmetry by Robert Gross, Avner Ash

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GALOIS THEORY 101
theorem” that we alluded to above. See chapter 14 for more on this
theorem.
Why Is G a Group?
Finally, we need to check that the permutations in G form a
group. That is, if g and h are in G, so that they permute the
algebraic numbers Q
alg
, then we must check that their composition
as permutations, g h, also permutes Q
alg
and satisfies the “rules of
the club” (8.3)–(8.6) on page 95. This is not hard to do. We also must
check that if g is in G, then its inverse permutation g
1
also abides
by the rules of the club—which is also not hard to do. Easiest of all,
we must check that the identity permutation e abides by these four
rules. In other words, if g and h are in G,soareg h, g
1
, and e.So
G is a group. (The associative property of the group law still holds,
because composition of permutations is always associative.)
Summary
We have covered an immense amount of ground in this chapter
in order to describe the Galois group, or symmetry group, of the
algebraic numbers. This is the key object we will be using as the
source of the representations we want to discuss later. Since it is so
key, here is a summary:
1. Q
alg
is the set of all complex numbers that can appear as
roots of Z-polynomials.
2. The Galois group G is the set of all permutations of Q
alg
that preserve the operations of addition and multiplication.
3. If g is any element of G and f (x)isanyZ-polynomial, then
as g acts as a permutation of Q
alg
, it permutes the roots of
f (x). It never maps some root of f (x) to a nonroot of f (x).
4. G has infinitely many elements.
102 CHAPTER 8
5. The only two elements of G that we can describe explicitly
in toto are e, the identity permutation and c, complex
conjugation.
6. Any element g of G can be partially described by taking a
Z-polynomial f (x), listing its roots a
1
, ..., a
n
, and telling
what permutation of these n algebraic numbers occurs
when we apply g.
7. Zorn’s Lemma plus some advanced algebra can be used to
piece together the partial descriptions of item 6 to get
elements of G. That is how we know G is infinite.

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