finite field

A finite fieldMathworldPlanetmath (also called a Galois field) is a field that has finitely many elements. The number of elements in a finite field is sometimes called the order of the field. We will present some basic facts about finite fields.

1 Size of a finite field

Theorem 1.1.

A finite field F has positive characteristic p>0 for some prime p. The cardinality of F is pn where n:=[F:Fp] and Fp denotes the prime subfieldMathworldPlanetmath of F.


The characteristicPlanetmathPlanetmath of F is positive because otherwise the additivePlanetmathPlanetmath subgroupMathworldPlanetmathPlanetmath generated by 1 would be an infinite subset of F. Accordingly, the prime subfield 𝔽p of F is isomorphicPlanetmathPlanetmathPlanetmath to the field /p of integers mod p. The integer p is prime since otherwise /p would have zero divisorsMathworldPlanetmath. Since the field F is an n–dimensional vector space over 𝔽p for some finite n, it is set–isomorphic to 𝔽pn and thus has cardinality pn. ∎

2 Existence of finite fields

Now that we know every finite field has pn elements, it is natural to ask which of these actually arise as cardinalities of finite fields. It turns out that for each prime p and each natural numberMathworldPlanetmath n, there is essentially exactly one finite field of size pn.

Lemma 2.1.

In any field F with m elements, the equation xm=x is satisfied by all elements x of F.


The result is clearly true if x=0. We may therefore assume x is not zero. By definition of field, the set F× of nonzero elements of F forms a group under multiplicationPlanetmathPlanetmath. This set has m-1 elements, and by Lagrange’s theoremMathworldPlanetmath xm-1=1 for any xF×, so xm=x follows. ∎

Theorem 2.2.

For each prime p>0 and each natural number nN, there exists a finite field of cardinality pn, and any two such are isomorphic.


For n=1, the finite field 𝔽p:=/p has p elements, and any two such are isomorphic by the map sending 1 to 1.

In general, the polynomialMathworldPlanetmathPlanetmathPlanetmath f(X):=Xpn-X𝔽p[X] has derivative -1 and thus is separablePlanetmathPlanetmath over 𝔽p. We claim that the splitting fieldMathworldPlanetmath F of this polynomial is a finite field of size pn. The field F certainly contains the set S of roots of f(X). However, the set S is closed underPlanetmathPlanetmath the field operations, so S is itself a field. Since splitting fields are minimalPlanetmathPlanetmath by definition, the containment SF means that S=F. Finally, S has pn elements since f(X) is separable, so F is a field of size pn.

For the uniqueness part, any other field F of size pn contains a subfieldMathworldPlanetmath isomorphic to 𝔽p. Moreover, F equals the splitting field of the polynomial Xpn-X over 𝔽p, since by Lemma 2.1 every element of F is a root of this polynomial, and all pn possible roots of the polynomial are accounted for in this way. By the uniqueness of splitting fields up to isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, the two fields F and F are isomorphic. ∎

Note: The proof of Theorem 2.2 given here, while standard because of its efficiency, relies on more abstract algebra than is strictly necessary. The reader may find a more concrete presentationMathworldPlanetmathPlanetmathPlanetmath of this and many other results about finite fields in [1, Ch. 7].

Corollary 2.3.

Every finite field F is a normal extensionMathworldPlanetmath of its prime subfield Fp.


This follows from the fact that field extensions obtained from splitting fields are normal extensions. ∎

3 Units in a finite field

Henceforth, in light of Theorem 2.2, we will write 𝔽q for the unique (up to isomorphism) finite field of cardinality q=pn. A fundamental step in the investigation of finite fields is the observation that their multiplicative groupsMathworldPlanetmath are cyclic:

Theorem 3.1.

The multiplicative group Fq* consisting of nonzero elements of the finite field Fq is a cyclic groupMathworldPlanetmath.


We begin with the formulaMathworldPlanetmathPlanetmath

dkϕ(d)=k, (1)

where ϕ denotes the Euler totient function. It is proved as follows. For every divisorMathworldPlanetmathPlanetmath d of k, the cyclic group Ck of size k has exactly one cyclic subgroup Cd of size d. Let Gd be the subset of Cd consisting of elements of Cd which have the maximum possible order ( of d. Since every element of Ck has maximal orderMathworldPlanetmath in the subgroup of Ck that it generates, we see that the sets Gd partitionPlanetmathPlanetmath the set Ck, so that


The identityPlanetmathPlanetmathPlanetmathPlanetmath (1) then follows from the observation that the cyclic subgroup Cd has exactly ϕ(d) elements of maximal order d.

We now prove the theorem. Let k=q-1, and for each divisor d of k, let ψ(d) be the number of elements of 𝔽q* of order d. We claim that ψ(d) is either zero or ϕ(d). Indeed, if it is nonzero, then let x𝔽q* be an element of order d, and let Gx be the subgroup of 𝔽q* generated by x. Then Gx has size d and every element of Gx is a root of the polynomial xd-1. But this polynomial cannot have more than d roots in a field, so every root of xd-1 must be an element of Gx. In particular, every element of order d must be in Gx already, and we see that Gx only has ϕ(d) elements of order d.

We have proved that ψ(d)ϕ(d) for all dq-1. If ψ(q-1) were 0, then we would have


which is impossible since the first sum must equal q-1 (because every element of 𝔽q* has order equal to some divisor d of q-1). ∎

A more constructive proofMathworldPlanetmath of Theorem 3.1, which actually exhibits a generator for the cyclic group, may be found in [2, Ch. 16].

Corollary 3.2.

Every extensionPlanetmathPlanetmathPlanetmath of finite fields is a primitive extension.


By Theorem 3.1, the multiplicative group of the extension field is cyclic. Any generator of the multiplicative group of the extension field also algebraically generates the extension field over the base fieldMathworldPlanetmath. ∎

4 Automorphisms of a finite field

Observe that, since a splitting field for Xqm-X over 𝔽p contains all the roots of Xq-X, it follows that the field 𝔽qm contains a subfield isomorphic to 𝔽q. We will show later (Theorem 4.2) that this is the only way that extensions of finite fields can arise. For now we will construct the Galois groupMathworldPlanetmath of the field extension 𝔽qm/𝔽q, which is normal by Corollary 2.3.

Theorem 4.1.

The Galois group of the field extension Fqm/Fq is a cyclic group of size m generated by the qth power Frobenius mapPlanetmathPlanetmath Frobq.


The fact that Frobq is an element of Gal(𝔽qm/𝔽q), and that (Frobq)m=Frobqm is the identity on 𝔽qm, is obvious. Since the extension 𝔽qm/𝔽q is normal and of degree m, the group Gal(𝔽qm/𝔽q) must have size m, and we will be done if we can show that (Frobq)k, for k=0,1,,m-1, are distinct elements of Gal(𝔽qm/𝔽q).

It is enough to show that none of (Frobq)k, for k=1,2,,m-1, is the identity map on 𝔽qm, for then we will have shown that Frobq is of order exactly equal to m. But, if any such (Frobq)k were the identity map, then the polynomial Xqk-X would have qm distinct roots in 𝔽qm, which is impossible in a field since qk<qm. ∎

We can now use the Galois correspondence between subgroups of the Galois group and intermediate fields of a field extension to immediately classify all the intermediate fields in the extension 𝔽qm/𝔽q.

Theorem 4.2.

The field extension Fqm/Fq contains exactly one intermediate field isomorphic to Fqd, for each divisor d of m, and no others. In particular, the subfields of Fpn are precisely the fields Fpd for dn.


By the fundamental theorem of Galois theory, each intermediate field of 𝔽qm/𝔽q corresponds to a subgroup of Gal(𝔽qm/𝔽q). The latter is a cyclic group of order m, so its subgroups are exactly the cyclic groups generated by (Frobq)d, one for each dm. The fixed field of (Frobq)d is the set of roots of Xqd-X, which forms a subfield of 𝔽qm isomorphic to 𝔽qd, so the result follows.

The subfields of 𝔽pn can be obtained by applying the above considerations to the extension 𝔽pn/𝔽p. ∎


  • 1 Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number TheoryMathworldPlanetmathPlanetmath, Second Edition, Springer–Verlag, 1990 (GTM 84).
  • 2 Ian Stewart, Galois Theory, Second Edition, Chapman & Hall, 1989.
Title finite field
Canonical name FiniteField
Date of creation 2013-03-22 12:37:50
Last modified on 2013-03-22 12:37:50
Owner yark (2760)
Last modified by yark (2760)
Numerical id 16
Author yark (2760)
Entry type Definition
Classification msc 12E20
Classification msc 11T99
Synonym Galois field
Related topic AlgebraicClosureOfAFiniteField
Related topic IrreduciblePolynomialsOverFiniteField