Compute Galois group when base field is not $\mathbb{Q}$.

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If i am correctly understanding what happens in the code, we use $$j=\sqrt {-1}\ ,\ a=\sqrt[4]{17}\ ,$$ and the tower of algebraic number fields $$\begin{array}{c} F=\Bbb Q(a,j) \\ \uparrow \\ k = \Bbb Q(j) \\ \uparrow \\ \Bbb Q \end{array}$$ and ask for the group $G=\text{Gal}(F:\Bbb Q)$, then check for each $g$ in $G$ if it stabilizes $j$.

This can be done by no longer using $k$ as follows:

R.<x> = PolynomialRing(QQ)
F.<j, a> = NumberField([x^2 + 1, x^4 - 17])
L.<w> = F.galois_closure()
G = L.galois_group()
J = (x^2 + 1).roots(ring=L, multiplicities=False)[0]
A = (x^4 - 17).roots(ring=L, multiplicities=False)[0]
Stab_J = [ g for g in G if g(J) == J ]

Note that working like this we obtain a list, not a subgroup of $G$.

sage: Stab_J
[(), (1,2,4,3)(5,7,8,6), (1,3,4,2)(5,6,8,7), (1,4)(2,3)(5,8)(6,7)]
sage: list(G)
[(),
 (1,2,4,3)(5,7,8,6),
 (1,3,4,2)(5,6,8,7),
 (1,4)(2,3)(5,8)(6,7),
 (1,5)(2,6)(3,7)(4,8),
 (1,6)(2,8)(3,5)(4,7),
 (1,7)(2,5)(3,8)(4,6),
 (1,8)(2,7)(3,6)(4,5)]
sage:

Here, we use $J,A$ - which are elements in the "abstract field" $L$, so that there are maps $j\to J$ and $a\to A$ from $F$ to $L$.