Find all c Z5 for which Z5xx3 2x c is a field Explain your

Find all c Z_5 for which Z_5[x]/(x^3 + 2x + c) is a field. Explain your reasoning.

Solution

Z5(x) / (x^3+2X+c)

This is a field if and only if p(x) = X^3 + 2X+ c

p(x) is irreducible if and only if it has no roots in Z5

taking different c values

p(0) = c

p(1) = 1+2+c = 3+c

p(2) = 8 + 4 + c = 12 + c

...............

Here for different values of p there exist some root depending on c

so Z5(x)/X^3+2x+c is a field

Here c may be any of the number

if c is a positive number then the given Z5(x) is a field

if it is a negative number depending on c value there may or may not exist a root and so that may or may not

be a field.