Answer to Question #183513 in Abstract Algebra for Anand

Question

Construct a field with 125 elements.

Answer 1

$Solution: ~We~ know ~ that \\To ~find ~ an ~ irreducible ~ polynomial ~ of ~ degree ~3 ~ in ~Z_5[x]. \\x^3 +x+1~ is ~ one ~ such ~polynomial~, it~ clearly ~has ~ no ~ linear~ factors~(Since ~0,1,2,3,4,) ~are ~\\not ~roots. \\So, F=\frac{Z_5[x]}{x^3 +x+1} ~is ~ a~ field ~ with ~ 5^3 ~elements.$