galois field

Example Sentences

• "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."