splitting field problem

[Dan Ismailescu]

The following quadratic polynomial

$P(X)= -51813033263*X^2+(-1291618080*sqrt(23)*sqrt(91)-7932964704*sqrt(91)*sqrt(2)-16045600662*sqrt(23)-13979137536*sqrt(2))*X+1551583008*sqrt(23)*sqrt(91)*sqrt(2)-29605395456*sqrt(23)*sqrt(2)-20092874592*sqrt(91)+95826938671$

can be factored as 

$(1/51813033263*(75686976*sqrt(97)*sqrt(91)*sqrt(23)-490070112*sqrt(2)*sqrt(97)*sqrt(91)-1444165632*sqrt(23)*sqrt(97)+1073509888*sqrt(2)*sqrt(97)+645809040*sqrt(23)*sqrt(91)+3966482352*sqrt(91)*sqrt(2)+8022800331*sqrt(23)+6989568768*sqrt(2)+51813033263*X))*(75686976*sqrt(97)*sqrt(91)*sqrt(23)-490070112*sqrt(2)*sqrt(97)*sqrt(91)-1444165632*sqrt(23)*sqrt(97)+1073509888*sqrt(2)*sqrt(97)-645809040*sqrt(23)*sqrt(91)-3966482352*sqrt(91)*sqrt(2)-8022800331*sqrt(23)-6989568768*sqrt(2)-51813033263*X)$

Note that the coefficients of $P$ are in $\mathbf{Q}[\sqrt{2},\sqrt{7},\sqrt{13},\sqrt{23}]$ while the roots are in $\mathbf{Q}[\sqrt{2},\sqrt{7},\sqrt{13},\sqrt{23},\sqrt{97}]$

How do I obtain this result using sage?

[Andrew]

Have you tried:

sage: RR.<X>=PolynomialRing(RR)
sage: p= -51813033263*X^2+(-1291618080*sqrt(23)*sqrt(91)-7932964704*sqrt(91)*sqrt(2)-16045600662*sqrt(23)-13979137536*sqrt(2))*X+1551583008*sqrt(23)*sqrt(91)*sqrt(2)-296053>
sage: p.roots()
[(-1968/51813033263*sqrt(91)*(328155*sqrt(23) + 2015489*sqrt(2)) - 8022800331/51813033263*sqrt(23) - 6989568768/51813033263*sqrt(2) - 1/103626066526*sqrt(3127640064*sqrt(91)*(195757394223*sqrt(23)*sqrt(2) - 884809888207) - 3822336404677065719808*sqrt(23)*sqrt(2) + 41118095321654931857408),
  1),
 (-1968/51813033263*sqrt(91)*(328155*sqrt(23) + 2015489*sqrt(2)) - 8022800331/51813033263*sqrt(23) - 6989568768/51813033263*sqrt(2) + 1/103626066526*sqrt(3127640064*sqrt(91)*(195757394223*sqrt(23)*sqrt(2) - 884809888207) - 3822336404677065719808*sqrt(23)*sqrt(2) + 41118095321654931857408),
  1)]