I am trying to solve two, 2-variable polynomial equations over F:=Q(i) modulo K:=F(√2). Specifically, if p1 = a2+6b2, p2 = 3a2+2b2, and $K^{4}:=\langle k^4\vert k\in K\backslash 0 \rangle,Iwanttofind(all?)aandbinFsuchthatp1==1moduloK^{4}andp2==−1moduloK^{*4}$.
Any amount of walk through or pointing in the right direction, or telling me this might not be doable would be great! I am relatively new to sage, or at least it has been years since I've used it.