# Difference between revisions of "Galois field"

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The number of elements of any finite field is a power $p^n$ of a prime number $p$, which is the [[Characteristic of a field|characteristic]] of this field. For any prime number $p$ and any natural number $n$ there exists a (unique up to an isomorphism) field of $p^n$ elements. It is denoted by $\mathrm{GF}(p^n)$ or by $\mathbb{F}_{p^n}$. The field $\mathrm{GF}(p^m)$ contains the field $\mathrm{GF}(p^n)$ as a subfield if and only if $m$ is divisible by $n$. In particular, any field $\mathrm{GF}(p^n)$ contains the field $\mathrm{GF}(p)$, which is called the [[prime field]] of characteristic $p$. The field $\mathrm{GF}(p)$ is isomorphic to the field $\mathbb{Z}/p\mathbb{Z}$ of residue classes of the ring of integers modulo $p$. In any fixed [[Algebraic closure|algebraic closure]] $\Omega$ of $\mathrm{GF}(p)$ there exists exactly one subfield $\mathrm{GF}(p^n)$ for each $n$. The correspondence $n \leftrightarrow \mathrm{GF}(p^n)$ is an isomorphism between the lattice of natural numbers with respect to division and the lattice of finite algebraic extensions (in $\Omega$) of $\mathrm{GF}(p)$ with respect to inclusion. The lattice of finite algebraic extensions of any Galois field within its fixed algebraic closure is such a lattice. | The number of elements of any finite field is a power $p^n$ of a prime number $p$, which is the [[Characteristic of a field|characteristic]] of this field. For any prime number $p$ and any natural number $n$ there exists a (unique up to an isomorphism) field of $p^n$ elements. It is denoted by $\mathrm{GF}(p^n)$ or by $\mathbb{F}_{p^n}$. The field $\mathrm{GF}(p^m)$ contains the field $\mathrm{GF}(p^n)$ as a subfield if and only if $m$ is divisible by $n$. In particular, any field $\mathrm{GF}(p^n)$ contains the field $\mathrm{GF}(p)$, which is called the [[prime field]] of characteristic $p$. The field $\mathrm{GF}(p)$ is isomorphic to the field $\mathbb{Z}/p\mathbb{Z}$ of residue classes of the ring of integers modulo $p$. In any fixed [[Algebraic closure|algebraic closure]] $\Omega$ of $\mathrm{GF}(p)$ there exists exactly one subfield $\mathrm{GF}(p^n)$ for each $n$. The correspondence $n \leftrightarrow \mathrm{GF}(p^n)$ is an isomorphism between the lattice of natural numbers with respect to division and the lattice of finite algebraic extensions (in $\Omega$) of $\mathrm{GF}(p)$ with respect to inclusion. The lattice of finite algebraic extensions of any Galois field within its fixed algebraic closure is such a lattice. | ||

− | The algebraic extension | + | The algebraic extension $\mathrm{GF}(p^n)/\mathrm{GF}(p)$ is simple, i.e. there exists a primitive element $\alpha \in \mathrm{GF}(p^n)$ such that $\mathrm{GF}(p^n) = \mathrm{GF}(p)(\alpha)$. Such an $\alpha$ will be any root of any irreducible polynomial of degree $n$ from the ring $\mathrm{GF}(p)[X]$. The number of primitive elements of the extension $\mathrm{GF}(p^n)/\mathrm{GF}(p)$ equals |

− | + | $$ | |

− | + | \sum_{d|n} \mu(d) p^{n/d} | |

− | + | $$ | |

− | where | + | where $\mu$ is the [[Möbius function|Möbius function]]. The additive group of the field $\mathrm{GF}(p^n)$ is naturally endowed with the structure of an $n$-dimensional vector space over $\mathrm{GF}(p)$. As a basis one may take $1,\alpha,\ldots,\alpha^{n-1}$. The non-zero elements of $\mathrm{GF}(p^n)$ form a multiplicative group, $\mathrm{GF}(p^n)^*$, of order $p^n-1$, i.e. each element of $\mathrm{GF}(p^n)^*$ is a root of the polynomial $X^{p^n-1}-1$. The group $\mathrm{GF}(p^n)^*$ is cyclic, and its generators are the primitive roots of unity of degree $p^n-1$, the number of which is $\phi(p^n-1)$, where $\phi$ is the [[Euler function|Euler function]]. Each primitive root of unity of degree $p^n-1$ is a primitive element of the extension $\mathrm{GF}(p^n)/\mathrm{GF}(p)$, but the converse is not true. More exactly, out of the |

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043140/g04314050.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043140/g04314050.png" /></td> </tr></table> |

## Revision as of 21:17, 11 October 2014

*finite field*

A field with a finite number of elements. First considered by E. Galois [1].

The number of elements of any finite field is a power $p^n$ of a prime number $p$, which is the characteristic of this field. For any prime number $p$ and any natural number $n$ there exists a (unique up to an isomorphism) field of $p^n$ elements. It is denoted by $\mathrm{GF}(p^n)$ or by $\mathbb{F}_{p^n}$. The field $\mathrm{GF}(p^m)$ contains the field $\mathrm{GF}(p^n)$ as a subfield if and only if $m$ is divisible by $n$. In particular, any field $\mathrm{GF}(p^n)$ contains the field $\mathrm{GF}(p)$, which is called the prime field of characteristic $p$. The field $\mathrm{GF}(p)$ is isomorphic to the field $\mathbb{Z}/p\mathbb{Z}$ of residue classes of the ring of integers modulo $p$. In any fixed algebraic closure $\Omega$ of $\mathrm{GF}(p)$ there exists exactly one subfield $\mathrm{GF}(p^n)$ for each $n$. The correspondence $n \leftrightarrow \mathrm{GF}(p^n)$ is an isomorphism between the lattice of natural numbers with respect to division and the lattice of finite algebraic extensions (in $\Omega$) of $\mathrm{GF}(p)$ with respect to inclusion. The lattice of finite algebraic extensions of any Galois field within its fixed algebraic closure is such a lattice.

The algebraic extension $\mathrm{GF}(p^n)/\mathrm{GF}(p)$ is simple, i.e. there exists a primitive element $\alpha \in \mathrm{GF}(p^n)$ such that $\mathrm{GF}(p^n) = \mathrm{GF}(p)(\alpha)$. Such an $\alpha$ will be any root of any irreducible polynomial of degree $n$ from the ring $\mathrm{GF}(p)[X]$. The number of primitive elements of the extension $\mathrm{GF}(p^n)/\mathrm{GF}(p)$ equals $$ \sum_{d|n} \mu(d) p^{n/d} $$ where $\mu$ is the Möbius function. The additive group of the field $\mathrm{GF}(p^n)$ is naturally endowed with the structure of an $n$-dimensional vector space over $\mathrm{GF}(p)$. As a basis one may take $1,\alpha,\ldots,\alpha^{n-1}$. The non-zero elements of $\mathrm{GF}(p^n)$ form a multiplicative group, $\mathrm{GF}(p^n)^*$, of order $p^n-1$, i.e. each element of $\mathrm{GF}(p^n)^*$ is a root of the polynomial $X^{p^n-1}-1$. The group $\mathrm{GF}(p^n)^*$ is cyclic, and its generators are the primitive roots of unity of degree $p^n-1$, the number of which is $\phi(p^n-1)$, where $\phi$ is the Euler function. Each primitive root of unity of degree $p^n-1$ is a primitive element of the extension $\mathrm{GF}(p^n)/\mathrm{GF}(p)$, but the converse is not true. More exactly, out of the

irreducible unitary polynomials of degree over there are polynomials of which the roots are generators of .

The set of elements of coincides with the set of roots of the polynomial in , i.e. is characterized as the subfield of elements from that are invariant with respect to the automorphism , which is known as the Frobenius automorphism. If , the extension is normal (cf. Extension of a field), and its Galois group is cyclic of order . The automorphism may be taken as the generator of .

#### References

[1] | E. Galois, "Écrits et mémoires d'E. Galois" , Gauthier-Villars (1962) |

[2] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |

[3] | N.G. [N.G. Chebotarev] Tschebotaröw, "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) (Translated from Russian) |

[4] | N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Springer (1989) pp. Chapt. 1–3 (Translated from French) |

**How to Cite This Entry:**

Galois field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Galois_field&oldid=30280