The Galois field GF(pⁿ) has order pⁿ and characteristic p.

The Galois field GF(pⁿ) is a finite extension of the Galois field GF(p) and the degree of the extension is n.

The multiplicative subgroup of a Galois field is cyclic.

A Galois field 𝔽_(pⁿ) is isomorphic to the quotient of the polynomial ring 𝔽ₚ adjoin x over the ideal generated by a monic irreducible polynomial of degree n. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: 𝔽_(pⁿ)≅𝔽ₚ[x]/(̂f_n(x)).

1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,
A field with a finite number of elements is called a Galois field.
The number of elements of the prime field k contained in a Galois field K is finite, and is therefore a natural prime p.

The case of most interest to us will be that in which F is a finite field, the Galois field GF(q) for some prime power q. If q is prime, this field is ℤ_q, the integers 0,1,…,q-1 with arithmetic modulo q.

2006, Debojyoti Battacharya, Debdeep Mukhopadhyay, D. RoyChowdhury, A Cellular Automata Based Approach for Generation of Large Primitive Polynomial and Its Application to RS-Coded MPSK Modulation, Samira El Yacoubi, Bastien Chopard, Stefania Bandini (editors), Cellular Automata: 7th International Conference, Proceedings, Springer, LNCS 4173, page 204,
Generation of large primitive polynomial over a Galois field has been a topic of intense research over the years. The problem of finding a primitive polynomial over a Galois field of a large degree is computationaly expensive and there is no deterministic algorithm for the same.

Galois field

Meaning

Noun