A finite field is a * field* with a finite

*(i.e., number of elements), also called a Galois field. The order of a finite field is always a*

**field order***or a*

**prime***of a*

**power***. For each*

**prime***, there exists*

**prime power***(with the usual caveat that “exactly one” means “exactly one up to an*

**exactly one***“) finite field GF(), often written as*

**isomorphism***in current usage.*

**F**_{pn}Theorem:

Let * E* be a finite field of characteristic

*.*

**p**1. The cardinality of * E *is

* |E| = p^{n}*, for some

*. It is denoted*

**n ≥ 1**

**E = F**_{pn}Furthermore, * E* is the splitting field for the separable polynomial

**f(X) = X**^{pn}− Xover * F_{p}*, so that any finite field with pn elements is isomorphic to

*. In fact,*

**E***coincides with the set of roots of*

**E***.*

**f**Proof:

1. Let * F_{p}* be the finite field with

*elements, given by the integers modulo*

**p***. Since*

**p***has characteristic*

**E***, it contains a copy of*

**p***. Thus*

**F**_{p}*is a field extension of*

**E***, and we may see*

**F**_{p}*as a vector space over*

**E***. If the dimension is*

**F**_{p}*, then let*

**n***be a basis. Every*

**α**_{1},…,α_{n}*in*

**x***can be written as*

**E****x = x _{1}α_{1} +···+ x_{n}α_{n}**

and there are * p* choices for each

*, thus a total of*

**x**_{i}*different elements in*

**p**^{n}*.*

**E**2. Let * E^{×}* be the multiplicative group of non-zero elements of

*. If*

**E***, then*

**α ∈ E**^{×}**αp ^{n}−1 = 1**

by Lagrange’s Theorem, so that

**αp ^{n} =α**

* ∀ α in E* (including

*). Thus each element of*

**α = 0***is a root off, and*

**E***is separable.*

**f**Now * f* has at most

*distinct roots, and we have already identified the*

**p**^{n}*elements of*

**p**^{n}*as roots of*

**E***.*

**f**Corollary: If * E* is a finite field of characteristic

*, then*

**p***is a Galois extension, with cyclic Galois group, generated by the Frobenius automorphism*

**E/F**_{p}**σ : x → σ(x) = x ^{p}, x ∈ E**

Proof:

By the above proposition, we know that * E* is a splitting field for a separable polynomial over

*, thus*

**F**_{p}*is Galois.*

**E/F**_{p}Since * x^{p} = x ∀ x in F_{p}*, we have that

**F _{p} ⊂ F(⟨σ⟩)**

that is * F_{p}* is contained in the fixed field of the cyclic subgroup generated by the Frobenius automorphism

*. But conversely, each element fixed by σ is a root of*

**σ***so*

**X**^{p}− X*has at most*

**F(⟨σ⟩)***elements. Consequently*

**p****F _{p} = F(⟨σ⟩)**

and

**Gal(E/F _{p}) = ⟨σ⟩**

This can be generalized when the base field is larger than * F_{p}*.

Corollary: Let * E/F* be a finite field extension with

*and*

**|E| = p**^{n}*. Then*

**|F| = p**^{m}*is a Galois extension and*

**E/F***. Furthermore, the Galois group is cyclic, generated by the automorphism*

**m|n****τ : x → τ(x) = x ^{pm}, x ∈ E**

Proof:

If the degree * [E : F] = d*, then every

*in*

**x***can be written as*

**E*** x = x_{1}α_{1} +···+ x_{d}α_{d}* and there are

*choices for each*

**p**^{m}*, thus a total of*

**x**_{i}* (p^{m})^{d} = p^{n}* different elements in

*, so that*

**E****d = m/n**

and

**m|n**

The same proof as for the above corollary holds for the rest.

Thus a way to construct a finite field * E* is, given

*and*

**p***, to construct*

**n***as a splitting field for*

**E = F**_{pn}*over*

**X**^{pn}− X

**F**_{p}Theorem:

If * G* is a finite subgroup of the multiplicative group of an arbitrary field, then

*is cyclic. Thus in particular, the multiplicative group*

**G***of a finite field*

**E**^{×}*is cyclic.*

**E**Proof:

The proof relies on the following fact: if * G* is a finite abelian group, it contains an element

*whose order*

**g***is the exponent of*

**r***, that is, the least common multiple of the orders of all elements of*

**G***.*

**G**Assuming this fact, we proceed as follows: if * x ∈ G*, then its order divides

*and thus*

**r****x ^{r} = 1**

Therefore each element of * G* is a root of

*and*

**X**^{r}− 1**|G| ≤ r**

Conversely, * |G|* is a multiple of the order of every element, so

*is at least as big as their least common multiple, that is*

**|G|**and

**|G| ≥ r |G| = r**

Since the order of * |G|* is

*, and it coincides with the order of the element g whose order is the exponent, we have that*

**r***is generated by*

**G***, that is*

**g***is cyclic. Since*

**G = ⟨g⟩***is cyclic, it is generated by a single element, say*

**E**^{×}*and*

**α : E = F**_{p}(α)*is called a primitive element of*

**α***. The minimal polynomial of*

**E***is called a primitive polynomial.*

**α**