# Math Help - Show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>

1. ## Show that x^3-2 is irreducible in Z/19Z [x]/<x^2-2>

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

2. Originally Posted by aznmaven
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 $t^3 - 2$ has no roots in $\mathbb{Z}_{19}[x]/(x^2-2)$. Let $\alpha = x + (x^2-2)$. Then we see that $\alpha^2 = 2$.

Now let $a+b\alpha$ be an element in this field with $a,b\in \mathbb{Z}_{19}$. Now show that $(a+b\alpha)^3 - 2 =0$ is impossible by expanding out this the expression and using the fact $\alpha^2 = 2$, and comparing coefficient.