Finite field
The most common examples of finite fields are the integers mod
The number of elements of a finite field is called its order or, sometimes, its size.
[1] Moreover, a field cannot contain two different finite subfields with the same order.
One may therefore identify all finite fields with the same order, and they are unambiguously denoted
The non-zero elements of a finite field form a multiplicative group.
(sometimes called the freshman's dream) is true in a field of characteristic
, which in general implies that the splitting field is a separable extension of the original).
In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:[1] The order of a finite field is a prime power.
The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic field extensions.
In the next sections, we will show how the general construction method outlined above works for small finite fields.
are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo
Every nonzero element of a finite field is a root of unity, as
consists of evenly spaced points around the unit circle (omitting zero).
has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with minimal polynomial of degree
This implies that, over GF(2), there are exactly 9 = 54/6 irreducible monic polynomials of degree 6.
Euler's totient function shows that there are 6 primitive 9th roots of unity,
The fact that the Frobenius map is surjective implies that every finite field is perfect.
They are a key step for factoring polynomials over the integers or the rational numbers.
As Xq − X does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it.
By the above formula, the number of irreducible (not necessarily monic) polynomials of degree n over GF(q) is (q − 1)N(q, n).
For every q and every n, the right hand side is positive, so there is at least one irreducible polynomial of degree n over GF(q).
For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field.
The finite field almost always has characteristic of 2, since computer data is stored in binary.
Some CPUs have special instructions that can be useful for finite fields of characteristic 2, generally variations of carry-less product.
For example, the fastest known algorithms for polynomial factorization and linear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then reconstruction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.
Many recent developments of algebraic geometry were motivated by the need to enlarge the power of these modular methods.
Wiles' proof of Fermat's Last Theorem is an example of a deep result involving many mathematical tools, including finite fields.
The Weil conjectures concern the number of points on algebraic varieties over finite fields and the theory has many applications including exponential and character sum estimates.
Finite fields have widespread application in combinatorics, two well known examples being the definition of Paley Graphs and the related construction for Hadamard Matrices.
It is not only unique up to an isomorphism, as do all algebraic closures, but contrarily to the general case, all its subfield are fixed by all its automorphisms, and it is also the algebraic closure of all finite fields of the same characteristic p. This property results mainly from the fact that the elements of
