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''finite field''

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

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 $\mathrm{GF}(p^n)/\mathrm{GF}(p)$ 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|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
$$
\frac{1}{n} \sum_{d|n} \mu(d) p^{n/d}
$$
irreducible unitary polynomials of degree $n$ over $\mathrm{GF}(p)$ there are $\phi(p^n-1)/n$ polynomials of which the roots are generators of $\mathrm{GF}(p^n)$.

The set of elements of $\mathrm{GF}(p^n)$ coincides with the set of roots of the polynomial $X^{p^n} - X$ in $\Omega$, i.e. $\mathrm{GF}(p^n)$ is characterized as the subfield of elements from $\Omega$ that are invariant with respect to the automorphism $\tau : x \mapsto x^{p^n}$, which is known as the Frobenius automorphism. If $\mathrm{GF}(p^m) \supset \mathrm{GF}(p^n)$, the extension $\mathrm{GF}(p^m)/\mathrm{GF}(p^n)$ is normal (cf. [[Extension of a field|Extension of a field]]), and its [[Galois group|Galois group]] $\mathrm{Gal}\left({\mathrm{GF}(p^m)/\mathrm{GF}(p^n)}\right)$ is cyclic of order $m/n$. The automorphism $\tau$ may be taken as the generator of $\mathrm{Gal}\left({\mathrm{GF}(p^m)/\mathrm{GF}(p^n)}\right)$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Galois,   "Écrits et mémoires d'E. Galois" , Gauthier-Villars  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. [N.G. Chebotarev] Tschebotaröw,   "Grundzüge der Galois'schen Theorie" , Noordhoff  (1950)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,   "Algebra" , ''Elements of mathematics'' , '''1''' , Springer  (1989)  pp. Chapt. 1–3  (Translated from French)</TD></TR></table>