Galois field

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.
• 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.