List of Finite Fields
Introduction
This article lists the finite fields that the corpus introduces — the prime fields $\mathbb{F}_p$ and the fields $\mathbb{F}_{p^n}$ of prime power order — with their orders, their constructions, their subfields, their Frobenius automorphism and their multiplicative group. Every entry points to the article that introduces the object.
The list is short because the theory is complete: for every prime power $q = p^n$ there is exactly one field of order $q$ up to isomorphism, and there are no others. The prime fields are the residue rings $\mathbb{Z}/p\mathbb{Z}$; the fields of composite prime power order are constructed as quotients $\mathbb{F}_p[x]/(f)$ by an irreducible polynomial of degree $n$; the subfields correspond to the divisors of $n$; the Frobenius generates the Galois group; and the multiplicative group is cyclic of order $q - 1$.
The article introduces nothing and proves nothing. It records examples and non-examples side by side.
The Prime Fields
| Field | Order | Construction | Subfields | Introduced in |
|---|---|---|---|---|
| $\mathbb{F}_2$ | $2$ | $\mathbb{Z}/2\mathbb{Z}$ | none | Finite Fields |
| $\mathbb{F}_3$ | $3$ | $\mathbb{Z}/3\mathbb{Z}$ | none | Finite Fields |
| $\mathbb{F}_5$ | $5$ | $\mathbb{Z}/5\mathbb{Z}$ | none | Finite Fields |
| $\mathbb{F}_p$ | $p$ | $\mathbb{Z}/p\mathbb{Z}$ | none: the prime field is minimal | Finite Fields |
A prime field has no proper subfield, and every field of characteristic $p$ contains a copy of $\mathbb{F}_p$ as its prime subfield. The residue ring $\mathbb{Z}/n\mathbb{Z}$ is a field exactly when $n$ is prime, by the criterion of Modular Arithmetic and the Ring of Residues.
The Fields of Prime Power Order
| Field | Order | Construction | Subfields | Introduced in |
|---|---|---|---|---|
| $\mathbb{F}_4$ | $4$ | $\mathbb{F}_2[x]/(x^2 + x + 1)$ | $\mathbb{F}_2$ | Finite Fields |
| $\mathbb{F}_8$ | $8$ | $\mathbb{F}_2[x]/(x^3 + x + 1)$ | $\mathbb{F}_2$ | Finite Fields |
| $\mathbb{F}_9$ | $9$ | $\mathbb{F}_3[x]/(x^2 + 1)$ | $\mathbb{F}_3$ | Finite Fields |
| $\mathbb{F}_{16}$ | $16$ | $\mathbb{F}_2[x]/(x^4 + x + 1)$ | $\mathbb{F}_2$, $\mathbb{F}_4$ | Finite Fields |
| $\mathbb{F}_{27}$ | $27$ | $\mathbb{F}_3[x]/(x^3 + 2x + 1)$ | $\mathbb{F}_3$ | Finite Fields |
| $\mathbb{F}_{p^n}$ | $q = p^n$ | $\mathbb{F}_p[x]/(f)$, $f$ irreducible of degree $n$ | the $\mathbb{F}_{p^m}$ for $m \mid n$ | Finite Fields |
| $\mathbb{F}_{p^m}$ | $p^m$ | the subfield fixed by the Frobenius of order $m$ | the divisors of $m$ | Finite Fields |
The construction depends on an irreducible polynomial, and a different irreducible polynomial of the same degree gives an isomorphic field; this is the uniqueness half of the classification. The additive group of $\mathbb{F}_{p^n}$ is the vector space $\mathbb{F}_p^n$, so the order is $p^n$ and the degree over the prime field is $n$.
The Subfield Lattice
| Field | Its subfields | The rule | Introduced in |
|---|---|---|---|
| $\mathbb{F}_{p^n}$ | $\mathbb{F}_{p^m}$ for each $m \mid n$ | the subfields correspond to the divisors of the degree | Finite Fields |
| $\mathbb{F}_{p^n}$ | no proper subfield when $n$ is prime | the only divisors are $1$ and $n$ | Finite Fields |
| $\mathbb{F}_{16}$ | $\mathbb{F}_2$ and $\mathbb{F}_4$ | $1 \mid 4$ and $2 \mid 4$ | Finite Fields |
| $\mathbb{F}_{p^n}$ | $\mathbb{F}_{p^m} \subseteq \mathbb{F}_{p^n}$ exactly when $m \mid n$ | the inclusion criterion | Finite Fields |
The lattice of subfields of $\mathbb{F}_{p^n}$ is the lattice of divisors of $n$ under divisibility, and this is the first example of the Galois correspondence. The extension $\mathbb{F}_{p^n}/\mathbb{F}_p$ is Galois and separable, with cyclic Galois group of order $n$.
The Frobenius and the Galois Group
| Object | Its action | Its order | Introduced in |
|---|---|---|---|
| the Frobenius $\varphi$ of $\mathbb{F}_{p^n}$ | $\varphi(x) = x^p$ | $n$ | Finite Fields |
| the Galois group $\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ | cyclic, generated by $\varphi$ | $n$, isomorphic to $\mathbb{Z}/n\mathbb{Z}$ | Finite Fields |
| the fixed field of $\varphi^m$ | $\mathbb{F}_{p^m}$ | the degree of the fixed field is $m$ | Finite Fields |
| the Frobenius on $\overline{\mathbb{F}_p}$ | $x \mapsto x^p$, an automorphism of infinite order | infinite | Finite Fields |
| the Frobenius $x \mapsto x^p$ on any characteristic-$p$ field | injective | the identity exactly on $\mathbb{F}_p$ | Fields |
The Frobenius is the identity on $\mathbb{F}_p$ and permutes the roots of an irreducible polynomial transitively; the fixed field of $\varphi^m$ is $\mathbb{F}_{p^m}$, which is the reason the subfield criterion is $m \mid n$. The same map on the algebraic closure $\overline{\mathbb{F}_p}$ has infinite order, and the finite fields generated by its orbits are the $\mathbb{F}_{p^n}$.
The Extensions Between Finite Fields
Every extension of finite fields is separable and Galois, which is why the subfield lattice is as simple as it is.
| Extension | Degree | Galois group | The Frobenius that generates it | Introduced in |
|---|---|---|---|---|
| $\mathbb{F}_{p^n}/\mathbb{F}_p$ | $n$ | $\mathbb{Z}/n\mathbb{Z}$ | $\varphi$ | Finite Fields |
| $\mathbb{F}_{p^n}/\mathbb{F}_{p^m}$, $m \mid n$ | $n/m$ | $\mathbb{Z}/(n/m)\mathbb{Z}$ | $\varphi^m$ | Finite Fields |
| $\mathbb{F}_{p^n}/\mathbb{F}_{p^n}$ | $1$ | trivial | $\varphi^n = \operatorname{id}$ | Finite Fields |
The trace and the norm of a finite extension are the two additive and multiplicative invariants of the extension, and for finite fields the trace is the sum of the Frobenius conjugates and the norm their product; both are recorded in Field Extensions. The extension is generated by a single element, since the multiplicative group is cyclic, so every finite field is a simple extension of every subfield.
The Multiplicative Group
| Field | The multiplicative group | Order | A generator | Introduced in |
|---|---|---|---|---|
| $\mathbb{F}_2$ | trivial | $1$ | $1$ | Finite Fields |
| $\mathbb{F}_3$ | $\{1, 2\}$ | $2$ | $2$ | Finite Fields |
| $\mathbb{F}_4$ | cyclic | $3$ | a root of $x^2 + x + 1$ | Finite Fields |
| $\mathbb{F}_8$ | cyclic | $7$ | a primitive root | Finite Fields |
| $\mathbb{F}_q$ | cyclic of order $q - 1$ | $q - 1$ | a primitive root | Finite Fields |
| $\overline{\mathbb{F}_p}$ | torsion, the union of the cyclic groups of order $p^n - 1$ | infinite | — | Finite Fields |
The multiplicative group of a finite field is cyclic, so a primitive root generates it, and the nonzero elements of $\mathbb{F}_q$ are exactly the roots of $x^{q-1} - 1$. The cyclotomic polynomial $\Phi_{q-1}$ therefore describes the primitive roots, and the situation parallels $\mathbb{Q}(\zeta_n)$ with $\zeta_n$ a primitive $n$-th root of unity in Cyclotomic Fields.
Warnings
| Object | Why it is not a finite field | Introduced in |
|---|---|---|
| $\mathbb{F}_p((t))$ | infinite: a Laurent series field | Absolute Values, Valuations and Completions |
| $\overline{\mathbb{F}_p}$ | infinite: the algebraic closure, the union of the finite fields | Finite Fields |
| $\mathbb{Z}/p^n\mathbb{Z}$, $n \geq 2$ | a finite ring of order $p^n$, but not a field: $p$ is a nilpotent | Reduced Rings and the Nilradical |
| $\mathbb{Z}/6\mathbb{Z}$ | a finite ring, not a field: $2 \cdot 3 = 0$ | Modular Arithmetic and the Ring of Residues |
| a finite division ring | by Wedderburn's little theorem it is a field, so there are no finite skew fields | Division Rings |
The last row is the reason the finite fields are exhausted by this list: a finite ring with no zero divisors is commutative by Wedderburn's little theorem, so the finite division rings are exactly the finite fields. The finite fields of characteristic $p$ are also the perfect fields, since the Frobenius is bijective on them.
Summary
This article has listed the finite fields of the corpus with their orders, constructions, subfields, Frobenius and multiplicative groups. The prime fields $\mathbb{F}_p$ are the residue rings $\mathbb{Z}/p\mathbb{Z}$ with no proper subfield; the fields $\mathbb{F}_{p^n}$ are the quotients by an irreducible polynomial of degree $n$, with subfields $\mathbb{F}_{p^m}$ exactly for $m \mid n$; the Frobenius $x \mapsto x^p$ generates the cyclic Galois group of order $n$ and has fixed field $\mathbb{F}_{p^m}$ for the power $m$; and the multiplicative group is cyclic of order $q - 1$. The infinite fields $\mathbb{F}_p((t))$ and $\overline{\mathbb{F}_p}$ and the finite non-fields $\mathbb{Z}/p^n\mathbb{Z}$ and $\mathbb{Z}/6\mathbb{Z}$ are recorded as the objects that fail the definition.
Summary of Notation
A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following.
| Symbol | Meaning |
|---|---|
| $\mathbb{F}_p$, $\mathbb{F}_q$, $\mathbb{F}_{p^n}$ | The finite field of order $p$, $q$ or $p^n$ |
| $\overline{\mathbb{F}_p}$ | The algebraic closure of $\mathbb{F}_p$ |
| $p$, $q$, $n$ | Prime, prime power $q = p^n$, degree |
| $\varphi$ | The Frobenius automorphism $x \mapsto x^p$ |
| $\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)$ | The cyclic Galois group of order $n$ |
| $\mathbb{F}_q^\times$ | The multiplicative group, cyclic of order $q - 1$ |
| $\mathbb{Z}/n\mathbb{Z}$ | The residue ring, a field for $n$ prime |
| $\Phi_{q-1}$ | The cyclotomic polynomial of the primitive roots |
Further Reading
- Rudolf Lidl and Harald Niederreiter, Finite Fields (Cambridge University Press, 2nd ed. 1997), for the classification, the subfield lattice and the multiplicative group.
- Serge Lang, Algebra (Springer, revised 3rd ed. 2002), for finite fields as Galois extensions with cyclic group generated by the Frobenius.
- Neal Koblitz, A Course in Number Theory and Cryptography (Springer, 2nd ed. 1994), for tables of the small finite fields and their primitive roots.