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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