Galois field

Named after French mathematician Évariste Galois (1811–1832).

Galois field (plural Galois fields)

1. (algebra) A finite field.

   The Galois field ${\displaystyle \mathrm {GF} (p^{n})}$ has order ${\displaystyle p^{n}}$ and characteristic ${\displaystyle p}$ .

   • 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 ${\displaystyle k}$ contained in a Galois field ${\displaystyle K}$ is finite, and is therefore a natural prime ${\displaystyle p}$ .

   • 2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory‎, retrieved 2016-05-05:

     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 ${\displaystyle \mathbb {Z} _{q}}$ , the integers ${\displaystyle 0,1,\dots ,q-1}$ 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^[sic] expensive and there is no deterministic algorithm for the same.

• For a given order, if a Galois field exists, it is unique, up to isomorphism.
• Generally denoted ${\displaystyle \mathrm {GF} (n)}$ (but sometimes ${\displaystyle \mathbb {F} _{n}}$ ), where ${\displaystyle n}$ 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 ${\displaystyle \mathbb {F} _{1}}$ or, more jocularly, ${\displaystyle \mathbb {F} _{un}}$ (pun intended).

• Finite field arithmetic on Wikipedia.Wikipedia
• Finite ring on Wikipedia.Wikipedia
• Finite group on Wikipedia.Wikipedia
• Field with one element on Wikipedia.Wikipedia
• Galois geometry on Wikipedia.Wikipedia
• Galois theory on Wikipedia.Wikipedia
• Hamming space on Wikipedia.Wikipedia
• Galois field on Encyclopedia of Mathematics
• Galois field structure on Encyclopedia of Mathematics
• Finite Field on Wolfram MathWorld