THE GALOIS GROUP OF A POLYNOMIAL 155

law of addition, where

ι → 0

σ → 1

τ → 2.

If H and K are isomorphic, then everything “grouplike” about

them is the same. If H is inﬁnite, K is inﬁnite, and vice versa. Or if

H has 43 elements, so does K and vice versa. If H is commutative,

so is K and vice versa. And on and on, for all the ﬁne structure you

can think of, as long as you can state it in terms of the group law.

The inverse Galois problem is the following problem: Given

any ﬁnite group H, can you ﬁnd a Z-polynomial f (x) with G(f )

isomorphic to H? If you can prove that this is always possible, give

us a call! It is an important unsolved problem. It is not easy to see

why it is so difﬁcult, but you can take this as an indication of just

how complicated these ﬁelds of the form Q(f ) can be.

This problem may also give you a little insight into what mathe-

maticians do. They do not just keep proving the same old theorems

over and over, or add up large columns of numbers. Some of them

work on the inverse Galois problem. This means trying to come up

with a general proof where, given any H, you can always ﬁnd an f (x)

as above. If that is too difﬁcult, they try to do it for certain H’s or

classes of H’s. Here there has been some success, and some very

complicated and large ﬁnite groups are known to be isomorphic

to Galois groups of the form G(f ). One of them is a huge group

known as the “Monster.” You will have to ﬁnd a book on group

theory to ﬁnd out more about what that is. Sufﬁce it to say that

it has 2

46

· 3

20

· 5

9

· 7

6

· 11

2

· 13

3

· 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71

elements!

Two More Things

Here is a difﬁcult and lengthy exercise:

EXERCISE: Let f (x) = (x

2

− 2)(x

2

− 3). The roots of f are

√

2,

−

√

2,

√

3, and −

√

3.

156 CHAPTER 13

Show that G(f ) has exactly four elements, ι, σ , τ , and ρ,

where ι is the identity permutation on the roots and the other

permutations are the following:

3

•

σ (±

√

2) =∓

√

2, σ (±

√

3) =±

√

3;

•

τ (±

√

2) =±

√

2, τ (±

√

3) =∓

√

3;

•

ρ(±

√

2) =∓

√

2, ρ(±

√

3) =∓

√

3.

SOLUTION: Here is a sketch: Let γ be an element of G(f ). We

must show that γ is one of ι, σ , τ ,orρ.Nowγ (

√

2) must be a

root of x

2

− 2, so γ (

√

2) =

√

2orγ (

√

2) =−

√

2. Similarly,

γ (

√

3) =

√

3orγ (

√

3) =−

√

3. Therefore, ι, σ , τ, and ρ are the

only possibilities for γ .

The harder part is to show that all four possibilities work.

First you have to identify Q(f ), which you can check is the set

of all numbers of the form a + b

√

2 + c

√

3 + d

√

6, where

a, b, c, and d are rational numbers. Then check that each of

ι, σ , τ , and ρ preserves addition and multiplication in Q(f ).

To get you started, this is how you show that σ preserves

addition: If s = a + b

√

2 + c

√

3 + d

√

6 and s

= a

+ b

√

2

+ c

√

3 + d

√

6, then σ(s) = a − b

√

2 + c

√

3 − d

√

6 and

σ (s

) = a

− b

√

2 + c

√

3 − d

√

6.

Now you check that σ(s + s

) = (a + a

) − (b + b

)

√

2 +

(c + c

)

√

3 − (d + d

)

√

6 = (a − b

√

2 + c

√

3 − d

√

6) +

(a

− b

√

2 + c

√

3 − d

√

6) = σ(s) + σ (s

).

By the way, you may have noticed that in all our examples, the

number of elements in G(f ) is the same as the number of rational

parameters it takes to describe Q(f ). This is always true, and is one

of the main theorems of Galois Theory.

3

The ∓, ± notation means that in any single equation, you can resolve ± as + or −,in

which case you must resolve ∓ as the opposite sign. For example, σ (±

√

2) =∓

√

2isa

concise way of writing two equations: σ (+

√

2) =−

√

2 and σ (−

√

2) =+

√

2.

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