Galois field

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Finite field

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Etymology

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

Noun

Galois field (plural Galois fields)

1. (algebra) A finite field; a field that contains a finite number of elements.

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

   The Galois field ${\displaystyle \mathrm {GF} (p^{n})}$ is a finite extension of the Galois field ${\displaystyle \mathrm {GF} (p)}$ and the degree of the extension is ${\displaystyle n}$.

   The multiplicative subgroup of a Galois field is cyclic.

   A Galois field ${\displaystyle \mathbb {F} _{p^{n}}}$ is isomorphic to the quotient of the polynomial ring ${\displaystyle \mathbb {F} _{p}}$ adjoin ${\displaystyle x}$ over the ideal generated by a monic irreducible polynomial of degree ${\displaystyle n}$. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: ${\displaystyle \mathbb {F} _{p^{n}}\cong {\mathbb {F} _{p} \over ({\hat {f}}_{n}(x))}}$.

   • 1958 , 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 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 ${\displaystyle \mathrm {GF} (n)}$ (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).

Hypernyms

• perfect field

Further reading

• 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