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 where is prime. The Galois field of order , unique up to isomorphism, is denoted or . ring Clearly every Galois field is in particular a Field of prime characteristic.

Construction and uniqueness

Let be a where and is prime. Then is a separable polynomial. Moreover, a field has precisely elements iff it is the splitting field of over , field whence follows uniqueness.¹

Proof

Let be the splitting field of over , and be the set of roots of . Since the formal derivative , we have , and thus is a separable polynomial of order , so . We show is a field, whence , since by definition is generated by over .

To this end, let , whence and , so using the Freshman’s dream

If ,

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

For the converse, let be a field such that . Then , so the multiplicative order of every element is a divisor of . Therefore

and we already have . Thus, has roots in , 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 and an irreducible polynomial of degree , then is a field of order . ring

Properties

Let . Then

1. is a perfect field, and consequently, irreducible polynomials in 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. ↩