Galois field
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English
Etymology
Noun
- Template:Lb A finite field; a field that contains a finite number of elements.
- 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 contained in a Galois field is finite, and is therefore a natural prime .
- Template:Quote-web
- 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, Template:W 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 computationalyTemplate:Sic expensive and there is no deterministic algorithm for the same.
- 1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,
Usage notes
- For a given order, if a Galois field exists, it is unique, up to isomorphism.
- Generally denoted (but sometimes ), where 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 or, more jocularly, (pun intended).