Construct a field with 125 elements.
Solution: We know thatTo find an irreducible polynomial of degree 3 in Z5[x].x3+x+1 is one such polynomial ,it clearly has no linear factors (Since 0,1,2,3,4,) are not roots.So,F=Z5[x]x3+x+1 is a field with 53 elements.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.Solution: We know thatTo find an irreducible polynomial of degree 3 in Z5[x].x3+x+1 is one such polynomial ,it clearly has no linear factors (Since 0,1,2,3,4,) are not roots.So,F=x3+x+1Z5[x] is a field with 53 elements.
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