5.1. Finite ¬elds

If K is a ¬nite ¬eld, then K is an Fp -vector space. Our ¬rst goal is to count the elements of

K. Here is a more general result.

Lemma 5.1. Let F be a ¬nite ¬eld with q elements and let V be an F -vector space. Then

dimF V < ∞ if and only if V is ¬nite in which case |V | = q dimF V .

Proof. If d = dimF V < ∞, then for a basis v1 , . . . , vd we can express each element v ∈ V

uniquely in the form v = t1 v1 + · · · + td vd , where t1 , . . . , td ∈ F . Clearly there are exactly q d

such expressions, so |V | = q d .

Conversely, if V is ¬nite then any basis has ¬nitely many elements and so dimF V < ∞.

Corollary 5.2. Let F be a ¬nite ¬eld and E/F an extension. Then E is ¬nite if and only

if E/F is ¬nite and then |E| = |F |[E:F ] .

Corollary 5.3. Let K be a ¬nite ¬eld. Then K/Fp is ¬nite and |K| = p[K:Fp ] .

Our next task is to show that for each power pd there is a ¬nite ¬eld with pd elements. We

d

start with the algebraic closure Fp of Fp and consider the polynomial ˜pd (X) = X p ’ X ∈

Fp [X]. Notice that ˜pd (X) = ’1, hence by Proposition 3.55 every root of ˜pd (X) in Fp is

simple. Therefore by Corollary 1.34 ˜pd (X) must have exactly pd distinct roots in Fp , say

0, u1 , . . . , upd ’1 . Then in Fp [X] we have

d

X p ’ X = X(X ’ u1 ) · · · (X ’ upd ’1 ),

and each root is separable over Fp . Let

F0d = {u ∈ Fpd : u = 0}.

Fpd = {u ∈ Fp : ˜pd (u) = 0} ⊆ Fp , p

d ’1

Notice that u ∈ F0d if and only if up = 1.

p

1, Fpd is a ¬nite sub¬eld of Fp with pd elements and

Proposition 5.4. For each d

F0d = F— . Furthermore, the extension Fpd /Fp is separable.

pd

p

Proof. If u, v ∈ Fpd then by the Idiot™s Binomial Theorem 1.11,

d d d d d

(u + v)p ’ (u + v) = (up + v p ) ’ (u + v) = (up ’ u) + (v p ’ v) = 0,

d d d

(uv)p ’ uv = up v p ’ uv = uv ’ uv = 0.

d d

Furthermore, if u = 0 then up ’1 = 1 and so u has multiplicative inverse up ’2 . Hence Fpd Fp .

Notice that Fp Fpd , so Fpd /Fp is a ¬nite extension. In any ¬eld the non-zero elements are

always invertible, hence F0d = F— .

pd

p

Fp is called the Galois ¬eld of order pd .

Definition 5.5. The ¬nite sub¬eld Fpd

59

60 5. GALOIS EXTENSIONS FOR FIELDS OF POSITIVE CHARACTERISTIC

Fpd is often denoted GF(pd ). Of course, Fp1 = GF(p1 ) = Fp . Notice also that [Fpd : Fp ] = d.

Proposition 5.6. Let d 1.

d d

Fp is the splitting sub¬eld for each of the polynomials X p ’ X and X p ’1 ’ 1

(i) Fpd

over Fp .

(ii) Fpd Fp is the unique sub¬eld with pd elements.

(iii) If K is any ¬eld with pd elements then there is an monomorphism K ’’ Fp with image

Fpd , hence K ∼ Fpd .

=

Proof. (i) As Fpd consists of exactly the roots of ˜pd (X) in Fp , it is the splitting sub¬eld.

d

The non-zero elements of Fpd are the roots of X p ’1 ’ 1, so Fpd is also the splitting sub¬eld for

this polynomial.

Fp have pd elements. Notice that the non-zero elements of F form a group K —

(ii) Let K

under multiplication. This group is abelian and has pd ’ 1 elements, so by Lagrange™s Theorem,

d d

each element u ∈ K — has order dividing pd ’ 1, therefore up ’1 = 1 and so up = u. But this

means every element of K is a root of ˜pd (X) and so K Fpd ; equality follows since these

sub¬elds both have pd elements.

(iii) Apply the Monomorphism Extension Theorem 3.49 for K = E = Fp and L = K.

It is worth noting the following consequence of this result and the construction of Fpd .

Corollary 5.7. Let K be a ¬nite ¬eld of characteristic p. Then K/Fp is separable.

Example 5.8. Consider the polynomial X 4 ’ X ∈ F2 [X]. By inspection, in the ring F2 [X]

we ¬nd that

X 4 ’ X = X 4 + X = X(X 3 + 1) = X(X + 1)(X 2 + X + 1).

Now X 2 + X + 1 has no root in F2 so it must be irreducible in F2 [X]. Its splitting ¬eld is a

quadratic extension F2 (w)/F2 where w is one of the roots of X 2 + X + 1, the other being w + 1

since the sum of the roots is the coe¬cient of X. This tells us that every element of F4 = F2 (w)

can be uniquely expressed in the form a + bw with a, b ∈ F2 . To calculate products we use the

fact that w2 = w + 1, so for a, b, c, d ∈ F2 we have

(a + bw)(c + dw) = ac + (ad + bc)w + bdw2 = (ac + bd) + (ad + bc + bd)w.

Example 5.9. Consider the polynomial X 9 ’ X ∈ F3 [X]. Let us ¬nd an irreducible poly-

nomial of degree 2 in F3 [X]. Notice that X 2 + 1 has no root in F3 , hence X 2 + 1 ∈ F3 [X] is

irreducible; so if u ∈ F3 is a root of X 2 + 1 then F3 (u)/F3 has degree 2 and F3 (u) = F9 . Every

element of F9 can be uniquely expressed in the form a + bu with a, b ∈ F3 . Multiplication is

carried out using the relation u2 = ’1 = 2.

By inspection, in the ring F3 [X] we ¬nd that

X 9 ’ X = X(X 8 ’ 1) = (X 3 ’ X)(X 2 + 1)(X 2 + X ’ 1)(X 2 ’ X ’ 1).

So X 2 + X ’ 1 and X 2 ’ X ’ 1 are also quadratic irreducibles in F3 [X]. We can ¬nd their roots

in F9 using the quadratic formula since in F3 we have 2’1 = (’1)’1 = ’1. The discriminant of

X 2 + X ’ 1 is

1 ’ 4(’1) = 5 = 2 = u2 ,

so its roots are (’1)(’1 ± u) = 1 ± u. Similarly, the discriminant of X 2 ’ X ’ 1 is

1 ’ 4(’1) = 5 = 2 = u2

and its roots are (’1)(1 ± u) = ’1 ± u. Then we have

F9 = F3 (u) = F3 (1 ± u) = F3 (’1 ± u).

There are two issues we can now clarify.

Proposition 5.10. Let Fpm and Fpn be two Galois ¬elds of characteristic p. Then Fpm

Fpn if and only if m | n.

5.1. FINITE FIELDS 61

Proof. If Fpm Fpn , then by Corollary 5.2,

pn = (pm )[Fpn :Fpm ] ,

so m | n.

m

1. Then for u ∈ Fpm we have up = u, so

If m | n, write n = km with k

n mk m m(k’1) m(k’1) m

up = up = (up )p = up = · · · = up = u.

Hence u ∈ Fpn and therefore Fpm Fpn .

This means that we can think of the Galois ¬elds Fpn as ordered by divisibility. Here is the

diagram of sub¬elds for Fp24 showing extensions with no intermediate subextensions.

Fp24

(5.1)

{{

{{

{{

{{

Fp8 Fp12

{{

{{

{{

{{

Fp4 Fp6

gg

{{ gg

{{ gg

{{ gg

{{

Fp2 Fp3

gg

{{

gg

{{

gg

{{

gg

{{

Fp

Theorem 5.11. The algebraic closure of Fp is the union of all the Galois ¬elds of charac-

teristic p,

Fp = Fpn .

n>1

Furthermore, each element u ∈ Fp is separable over Fp .

Proof. Let u ∈ Fp . Then u is algebraic over Fp and the extension Fp (u)/Fp is ¬nite. Hence

by Corollary 5.2, Fp (u) Fp is a ¬nite sub¬eld. Proposition 5.10 now implies that Fp (u) = Fpn

for some n. The separability statement follows from Corollary 5.7.

We will require a useful fact about Galois ¬elds.

Proposition 5.12. The group of units F— in Fpd is cyclic.

pd

This is a special case of a more general result about arbitrary ¬elds.

K — is cyclic.

Proposition 5.13. Let K be a ¬eld. Then every ¬nite subgroup U

Proof. Use Corollary 1.34 and Lemma 1.45.

Definition 5.14. w ∈ F— is called a primitive root if it is a primitive (pd ’ 1)-th root of

pd

unity, i.e., its order in the group F— is (pd ’ 1), hence w = F— .

pd pd

Remark 5.15. Unfortunately the word primitive has two confusingly similar uses in the

context of ¬nite ¬elds. Indeed, some authors use the term primitive element for what we have

called a primitive root, but that con¬‚icts with our usage, although as we will in the next result,

every primitive root is indeed a primitive element in our sense!

Proposition 5.16. The extension of Galois ¬elds Fpnd /Fpd is simple, i.e., Fpnd = Fpd (u)

for some u ∈ Fpnd .

Proof. By Proposition 5.12, Fpnd has a primitive root w say. Then every element of Fpnd

is a polynomial in w, so Fpnd Fpd (w) Fpnd , hence Fpnd = Fpd (w).

62 5. GALOIS EXTENSIONS FOR FIELDS OF POSITIVE CHARACTERISTIC

Remark 5.17. This completes the proof of the Primitive Element Theorem 3.75 which we

had previously only established for in¬nite ¬elds.

Example 5.18. In Example 5.8 we ¬nd that F4 = F2 (w) has the two primitive roots w and

w + 1.

Example 5.19. In Example 5.9 we have F9 = F3 (u) and F— is cyclic of order 8. Since

9

•(8) = 4, there are four primitive roots and these are the roots of the polynomials X 2 + X ’ 1

and X 2 ’ X ’ 1 which we found to be ±1 ± u.

We record a fact that is very important in Number Theory.

Proposition 5.20. Let p > 0 be an odd prime.

(i) If p ≡ 1 (mod 4), the polynomial X 2 + 1 ∈ Fp [X] has two roots in Fp .

(ii) If p ≡ 3 (mod 4) the polynomial X 2 +1 ∈ Fp [X] is irreducible, so Fp2 ∼ Fp [X]/(X 2 +1).

=

Proof. (i) We have 4 | (p ’ 1) = |F— |, so if u ∈ F— is a generator of this cyclic group, the

p p

— —

order of u|Fp |/4 is 4, hence this is a root of X 2 + 1 (the other root is ’u|Fp |/4 ).

(ii) If v ∈ Fp is a root of X 2 + 1 then v has order 4 in F— . But then 4 | (p ’ 1) = |F— |, which is

p p

impossible since p ’ 1 ≡ 2 (mod 4).

Here is a generalization of Proposition 5.20.

Proposition 5.21. Fpd contains a primitive n-th root of unity if and only if pd ≡ 1 (mod n)

and p n.

5.2. Galois groups of ¬nite ¬elds and Frobenius mappings

Consider an extension of Galois ¬elds Fpnd /Fpd . By Proposition 5.6(i), Corollary 5.7 and

Proposition 3.73, this extension is Galois and

| Gal(Fpnd /Fpd )| = [Fpnd : Fpd ] = n.

We next introduce an important element of the Galois group Gal(Fpnd /Fpd ).

Definition 5.22. The (relative) Frobenius mapping for the extension Fpnd /Fpd is the func-

d

tion Fd : Fpnd ’’ Fpnd given by Fd (t) = tp .

Proposition 5.23. The relative Frobenius mapping Fd : Fpnd ’’ Fpnd is an automor-

phism of Fpnd that ¬xes the elements of Fpd , so Fd ∈ Gal(Fpnd /Fpd ). The order of Fd is n,

so Gal(Fpnd /Fpd ) = Fd , the cyclic group generated by Fd .

Proof. For u, v ∈ Fpnd , we have the identities

d d d d d d

Fd (u + v) = (u + v)p = up + v p , Fd (uv) = (uv)p = up v p ,

so Fd is a ring homomorphism. Also, for u ∈ Fpd we have

d

Fd (u) = up = u,

so Fd ¬xes the elements of Fpd . To see that Fd is an automorphism, notice that the composition

power Fn = Fd —¦ · · · —¦ Fd (with n factors) satis¬es

d

nd

Fn (t) = tp =t

d

for all t ∈ Fpnd , hence Fn = id. Then Fd is invertible with inverse F’1 = Fn’1 . This also shows

d d d

that the order of Fd in the group AutFpd (Fpnd ) is at most n. Suppose the order is k with k n;

kd

then every element u ∈ Fpnd satis¬es the equation Fk (u) = u which expands to up = u, hence

d

u ∈ Fpkd . But this can only be true if k = n.

Frobenius mappings exist on the algebraic closure Fp . For d 1, consider the function

d

Fd (t) = tp .

Fd : Fp ’’ Fp ;

5.2. GALOIS GROUPS OF FINITE FIELDS AND FROBENIUS MAPPINGS 63

Proposition 5.24. Let d 1.

(i) Fd : Fp ’’ Fp is an automorphism of Fp which ¬xes the elements of Fpd . In fact for

u ∈ Fp , Fd (u) = u if and only if u ∈ Fpd .

(ii) The restriction of Fd to the Galois sub¬eld Fpdn agrees with the relative Frobenius

mapping Fd : Fpnd ’’ Fpnd .

(ii) If k 1, then Fk = Fkd , so Fd has in¬nite order in the automorphism group AutFpd (Fp ),

d

hence this group is in¬nite.

Proof. This is left as an exercise.

The Frobenius mapping F = F1 is often called the absolute Frobenius mapping since it exists

as an element of each of the groups AutFp (Fp ) and AutFp (Fpn ) = Gal(Fpn /Fp ) for every n 1.

In Gal(Fpnd /Fpd ) = Fd , for each k with k | n there is the cyclic subgroup Fk of order

d

k

| Fd | = n/k.

Fk

Fk d

Proposition 5.25. For k | n, the ¬xed sub¬eld of in Fpnd is Fpnd = Fpdk .

d

Fpnd

n/k

Fk

d

Fpnd = Fpdk

k

Fpd

dk

Proof. For u ∈ Fpnd we have Fk (u) = up , hence Fk (u) = u if and only if u ∈ Fpdk .

d d

Here is the subgroup diagram corresponding to the lattice of sub¬elds of Fp24 shown in (5.1).

64 5. GALOIS EXTENSIONS FOR FIELDS OF POSITIVE CHARACTERISTIC