Finite field ( $GF (n)$ )

A convenient notation for the elements of

GF (n)

of prime order is to use integers, e.g. for

GF (7)

we could write:

GR (7) = {- 3, - 2, - 1, 0, 1, 2, 3}

(1)

which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used:

GR (7) = {0, 1, 2, 3, 4, 5, 6}

(2)

For fields of prime order, regular modular arithmetic works as the field operation.

For non-prime order, we see that modular arithmetic does not work because the divisors have no inverse. E.g. at order 6, 2 and 3 have no inverse, e.g. for 2:

0 \times 2 = 01 \times 2 = 22 \times 2 = 43 \times 2 = 04 \times 2 = 25 \times 2 = 4

(3)

we see that things wrap around perfecly, and 1 is never reached.

For non-prime prime power orders however, we can find a way, see finite field of non-prime order.

Video 1.

Finite fields made easy by Randell Heyman (2015)

Source. Good introduction with examples

Table of contents

Classification of finite fields

 0  0

There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as

GF (n)

Every element of a finite field satisfies

x^{or d er} = x

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

Finite field of non-prime order

 0  0

As per classification of finite fields those must be of prime power order.

Video "Finite fields made easy by Randell Heyman (2015)" at youtu.be/z9bTzjy4SCg?t=159 shows how for order

9 = 3 \times 3

. Basically, for order

p^{n}

, we take:

each element is a polynomial in $GF (p) [x]$ , $GF (p) [x]$ , the polynomial ring over the finite field $GF (p)$ with degree smaller than $n$ . We've just seen how to construct $GF (p)$ for prime $p$ above, so we're good there.
addition works element-wise modulo on $GF (p)$
multiplication is done modulo an irreducible polynomial of order $n$

For a worked out example, see: GF(4).

GF(2)

 1  0

GF(4)

 0  0

Ciro Santilli tried to add this example to Wikipedia, but it was reverted, so here we are, see also: Section "Deletionism on Wikipedia".

This is a good first example of a field of a finite field of non-prime order, this one is a prime power order instead.

4 = 2^{2}

, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):

0X + 0
0X + 1
1X + 0
1X + 1

Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of prime order such as GF(2), because we are dealing with prime powers only. And we have already defined fields of prime order easily previously with modular arithmetic.

Over GF(2), there is only one irreducible polynomial of degree 2:

X^{2} + X + 1

(1)

Addition is defined element-wise with modular arithmetic modulo 2 as defined over GF(2), e.g.:

(1 X + 0) + (1 X + 1) = (1 + 1) X + (0 + 1) = 0 X + 1

(2)

Multiplication is done modulo

X^{2} + X + 1

, which ensures that the result is also of degree 1.

For example first we do a regular multiplication:

(1 X + 0) \times (1 X + 1) = (1 \times 1) X^{2} + (1 \times 1) X + (0 \times 1) X + (0 \times 1) = 1 X^{2} + 1 X + 0

(3)

Without modulo, that would not be one of the elements of the field anymore due to the

1 X^{2}

So we take the modulo, we note that:

1 X^{2} + 1 X + 0 = 1 (X^{2} + X + 1) + (0 X + 1)

(4)

and by the definition of modulo:

(1 X^{2} + 1 X + 0) mod (X^{2} + X + 1) = (0 X + 1)

(5)

which is the final result of the multiplication.

TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.

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Finite field by

Codex  0 Created 2026-09-24 Updated 2026-10-13

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A finite field is a field with finitely many elements: addition, subtraction, multiplication, and division by a nonzero element all stay inside a finite collection. Write

F_{q}

, or

GF (q)

, for a finite field with

q

elements. The subscript counts elements; it does not prescribe modular arithmetic modulo

q

. The field with four elements is the smallest example where that distinction matters.

Every finite field has size

q = p^{n}

, where

p

is a prime number and

n

is a positive integer. To see the counting mechanism, use its prime field

F_{p}

: the finite field is a finite-dimensional vector space over

F_{p}

. For a basis

e_{1}, \dots, e_{n}

, every element has exactly one expression

c_{1} e_{1} + \dots + c_{n} e_{n}, c_{i} \in F_{p} .

(1)

Each coefficient has

p

choices, so there are

p^{n}

expressions. Here

p

is the characteristic of a field and

n

is the dimension of a vector space. Thus sizes

4, 8, 9, 16, 25

are possible, whereas

6, 10, 12

are not prime powers and cannot be finite field sizes.

Conversely, every prime power occurs, and any two finite fields of the same size admit a field isomorphism. These are classification statements, beyond the counting argument. See Conrad's Theorems 1.5, 2.2, and 2.7.^[ref] Uniqueness means the arithmetic agrees after relabeling; it does not choose a basis, an irreducible polynomial for a quotient ring representation, or a particular field isomorphism. Even for the field with four elements, exchanging

α

and

α + 1

preserves the arithmetic.

The appeal for coding theory is exact, finite arithmetic with reliable division: polynomial interpolation and linear algebra work over a finite alphabet. A Reed-Solomon code represents messages by polynomials and adds redundancy by evaluating them at several distinct finite field elements. Forney's MIT notes develop this arithmetic as preparation for such error-correcting codes.^[ref] In cryptography, finite fields likewise supply exact arithmetic on finitely many symbols, with operations that can be implemented using bits.

Historically, Évariste Galois published work on finite fields beyond prime fields in 1830, constructing them by adjoining elements satisfying irreducible polynomials over prime fields. Conrad's historical account also records E. H. Moore's 1893 terminology, from which the name Galois field and notation

GF (q)

descend.^[ref]

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Finite field ( $GF (n)$ )

Classification of finite fields

Finite field of non-prime order

GF(2)

GF(4)

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