Field

Galois field

A Galois field is a field containing a finite number of elements. ring The cardinality of a field is called its order, and finite fields only exist for orders of the form 𝑝ℎ where 𝑝 is prime. The Galois field of order 𝑝ℎ, unique up to isomorphism, is denoted 𝔽𝑝ℎ or GF(𝑝ℎ). ring Clearly every Galois field is in particular a Field of prime characteristic.

Construction and uniqueness

Let 𝑞 =𝑝ℎ be a where ℎ,𝑝 ∈ℕ and ℎ is prime. Then 𝑥𝑞 −𝑥 ∈𝔽𝑝[𝑥] is a separable polynomial. Moreover, a field 𝐹 has precisely 𝑞 elements iff it is the splitting field of 𝑥𝑞 −𝑥 over 𝔽𝑝, field whence follows uniqueness.¹

Proof

Let 𝐹 be the splitting field of 𝑥𝑞 −𝑥 over 𝕂𝑝, and 𝐸 ⊆𝐹 be the set of roots of 𝑥𝑞 −𝑥. Since the formal derivative 𝑓′(𝑥) =𝑞𝑥𝑞−1 −1 = −1, we have gcd{𝑓(𝑥),𝑓′(𝑥)} =1, and thus 𝑓(𝑥) is a separable polynomial of order 𝑞, so |𝐸| =𝑞. We show 𝐸 is a field, whence 𝐸 =𝐹, since by definition 𝐹 is generated by 𝐸 over 𝐾.

To this end, let 𝑎,𝑏 ∈𝐸, whence 𝑎𝑞 =𝑎 and 𝑏𝑞 =𝑏, so using the Freshman’s dream

(𝑎−𝑏)𝑝ℎ=𝑎𝑞+(−1)𝑞𝑏𝑞=𝑎𝑞−𝑏𝑞=𝑎−𝑏.

If 𝑏 ≠0,

(𝑎𝑏−1)𝑞=𝑎𝑞(𝑏𝑞)−1=𝑎𝑏−1

proving 𝐸 is closed under subtraction and division, thus it indeed a subfield by the Tests for subfields.

For the converse, let 𝐹 be a field such that |𝐹| =𝑞. Then |𝐹×| =𝑞 −1, so the multiplicative order of every element is a divisor of 𝑞 −1. Therefore

𝑎≠0⟹𝑎𝑞−1=1⟹𝑎𝑞−𝑎=0;

and we already have 0𝑞 −0 =0. Thus, 𝑥𝑞 −𝑥 has 𝑞 roots in 𝐹, whence it is the splitting field, as stated.

Direct construction as quotient by a polynomial

A finite field of a given order can be constructed as a quotient of a polynomial ring. Given a polynomial ring ℤ𝑝[𝑥] and an irreducible polynomial 𝑓 of degree ℎ, then ℤ𝑝[𝑥]/⟨𝑓⟩ is a field of order 𝑝ℎ. ring

Properties

Let 𝐾 =GF⁡(𝑝ℎ). Then

1. 𝐾 is a perfect field, and consequently, irreducible polynomials in 𝐾[𝑥] are separable.

Proof of 1

Since the Frobenius endomorphism is injective (Field homomorphisms are injective), by the Pigeonhole principle it must also be surjective, proving ^P1.

Other results

• Finite extension of a Galois field
• By Wedderburn’s little theorem these are the only finite division rings.

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Footnotes

1. 2009. Algebra: Chapter 0, §VII.5.1, p. 441. ↩