View source for Galois field

==English==
{{wikipedia|Finite field}}

===Etymology===
{{named-after|en|nat=French|occ=mathematician|Évariste Galois|wplink==|born=1811|died=1832}}.

===Noun===
{{en-noun}}

# {{lb|en|algebra}} A [[finite field]]; a field that contains a [[finite]] number of [[elements]].
#: {{ux|en|The '''Galois field''' <math>\mathrm{GF}(p^n)</math> has order <math>p^n</math> and [[characteristic]] <math>p</math>.}}
#: {{ux|en|The '''Galois field''' <math>\mathrm{GF}(p^n)</math> is a finite extension of the '''Galois field''' <math>\mathrm{GF}
(p)</math> and the degree of the extension is <math>n</math>.}}
#: {{ux|en|The multiplicative subgroup of a '''Galois field''' is cyclic.}}
#: {{ux|en|A '''Galois field''' <math>\mathbb{F}_{p^n}</math> is isomorphic to the quotient of the polynomial ring <math>\mathbb{F}_p</math> adjoin <math>x</math> over the ideal generated by a monic irreducible polynomial of degree <math>n</math>. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: <math>\mathbb{F}_{p^n} \cong {\mathbb{F}_p[x] \over (\hat f_n(x))} </math>.}}
#* '''1958''' [Chelsea Publishing Company], Hans J. Zassenhaus, ''The Theory of Groups'', 2013, Dover, [https://books.google.com.au/books?id=5TaoAAAAQBAJ&pg=PT120&dq=%22Galois+field%22%7C%22Galois+fields%22&hl=en&sa=X&ved=0ahUKEwjf_o-B4t_eAhWGkJAKHQw9BrsQ6AEIkgEwFQ#v=onepage&q=%22Galois%20field%22%7C%22Galois%20fields%22&f=false unnumbered page],
#*: A field with a finite number of elements is called a '''''Galois field'''''.
#*: The number of elements of the prime field <math>k</math> contained in a '''Galois field''' <math>K</math> is finite, and is therefore a natural prime <math>p</math>.
#* {{quote-web|en|author=[[w:http://home.gwu.edu/~jbonin/|Joseph E. Bonin]]|title=|work=A Brief Introduction To Matroid Theory|url=http://home.gwu.edu/~jbonin/survey.ps|archiveurl=|archivedate=|location=|publisher=|year=2001|accessdate=2016-05-05|passage=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 <math>\mathbb{Z}_q</math>, the integers <math>0, 1,\dots, q-1</math> 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, {{w|Lecture Notes in Computer Science|LNCS}} 4173, [https://books.google.com.au/books?id=_-G_44qVki4C&pg=PA204&dq=%22Galois+field%22%7C%22Galois+fields%22&hl=en&sa=X&ved=0ahUKEwjf_o-B4t_eAhWGkJAKHQw9BrsQ6AEIezAQ#v=onepage&q=%22Galois%20field%22%7C%22Galois%20fields%22&f=false 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.

====Usage notes====
* For a given order, if a Galois field exists, it is unique, up to [[isomorphism]].
* Generally denoted <math>\mathrm{GF}(n)</math> (but sometimes <math>\mathbb F_n</math>), where <math>n</math> 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 <math>\mathbb{F}_1</math> or, more jocularly, <math>\mathbb{F}_{un}</math> (pun intended).

====Hypernyms====
* {{l|en|perfect field}}

===Further reading===
* {{pedia|Finite field arithmetic}}
* {{pedia|Finite ring}}
* {{pedia|Finite group}}
* {{pedia|Field with one element}}
* {{pedia|Galois geometry}}
* {{pedia|Galois theory}}
* {{pedia|Hamming space}}
* [https://www.encyclopediaofmath.org/index.php/Galois_field Galois field] on {{w|Encyclopedia of Mathematics}}
* [https://www.encyclopediaofmath.org/index.php/Galois_field_structure  Galois field structure] on {{w|Encyclopedia of Mathematics}}
* [http://mathworld.wolfram.com/FiniteField.html Finite Field] on {{w|MathWorld|Wolfram MathWorld}}

{{cln|en|eponyms}}