Meaning of Galois field | Babel Free

Definitions

A finite field; a field that contains a finite number of elements.

Equivalents

Examples

“The Galois field #92;mathrm#123;GF#125;(pⁿ) has order pⁿ and characteristic p.”

“The Galois field #92;mathrm#123;GF#125;(pⁿ) is a finite extension of the Galois field #92;mathrm#123;GF#125;(p) and the degree of the extension is n.”

“The multiplicative subgroup of a Galois field is cyclic.”

“A Galois field #92;mathbb#123;F#125;#95;#123;pⁿ#125; is isomorphic to the quotient of the polynomial ring #92;mathbb#123;F#125;#95;p adjoin x over the ideal generated by a monic irreducible polynomial of degree n. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: #92;mathbb#123;F#125;#95;#123;pⁿ#125;#92;cong#123;#92;mathbb#123;F#125;#95;p#91;x#93;#92;over(#92;hatf#95;n(x))#125;.”

“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.”

“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 #92;mathbb#123;Z#125;#95;q, the integers 0,1,#92;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.”

CEFR level