Factorisaing ideals in quadratic number fields

Question

Show there is an ideal $a$ in $\mathbb{Z}[\sqrt-29]$ satisfying the equality $\langle8\rangle=a^{2}$.

I tried to factorise the minimal polynomial over $\mathbb{F}_{8}$ but it does'nt seem to work, what is the best approach to this question?

Answer 1

Hint: $8=2^3$

I assume you were factorising $X^2+29$ in $\mathbb{F}_8$, and trying to apply Dedekind's criterion. If this is the case, look closely at the conditions that must be satisfied in order to apply Dedekins criterion, and then you should see why it didn't work.