Galois field

Etymology
.

Noun

 * 1)  A finite field; a field that contains a finite number of elements.
 * 2) * 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 $$p$$.
 * 1) * 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, 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 [sic] expensive and there is no deterministic algorithm for the same.
 * 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 $$p$$.
 * 1) * 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, 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 [sic] expensive and there is no deterministic algorithm for the same.
 * 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 [sic] expensive and there is no deterministic algorithm for the same.

Usage notes

 * For a given order, if a Galois field exists, it is unique, up to isomorphism.
 * Generally denoted $$\mathrm{GF}(n)$$ (but sometimes $$\mathbb F_n$$), where $$n$$ is the number of elements, which must be a positive integer power of a prime.
 * Although, strictly speaking, the "field of one element" does not exist (it is not a field in classical algebra), it is occasionally discussed in terms of how it might be meaningfully defined. Were it a meaningful concept, it would be a Galois field. It may be denoted $$\mathbb{F}_1$$ or, more jocularly, $$\mathbb{F}_{un}$$ (pun intended).