Please show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>. There HAS to be a better way than plugging and chugging 361 polynomials.
Thanks!
Julian
It is sufficient to show has no roots in . Let . Then we see that .
Now let be an element in this field with . Now show that is impossible by expanding out this the expression and using the fact , and comparing coefficient.