Question

Briefly describe the main properties of Galois Field?

Answer 1

1) Let F be a Galois field containing a subfield K with q elements. Then F has qm elements, where m=[F :K].

2)Let F be a Galois field. Then F has pn elements, where the prime p is the characteristic of F and n is the degree of F over its prime subfield.

3) (Existence and Uniqueness of Finite Fields)

For every prime p and every positive integer n, there exists a Galois field with pn elements. Any F inite field with q = pn elements is isomorphic to the splitting field of xq − x over Fp.

4)For every Galois field Fq, the multiplicative group F*q of nonzero elements of Fq is cyclic.

5)For every Galois field Fq and every positive integer n, there exists an irreducible polynomial in Fq[x] of degree n.