This is my current code:

with~([NumberTheory, DocumentTools]):
tb := Tabulate([seq]([galois](CyclotomicPolynomial(i, x),x), i = 1 .. 10)):

I hope to export it into SVG file, but this code will report a error information:

Export("aatest.svg", tb)

As the code:

poly := x^4 + 8*x + 12:
galois(poly, x)

"4T4", {"A(4)"}, "+", 12, {"(1 2 4)", "(2 3 4)"}

 Then I know it's Galois group has to be (isomorphic to) A4. And I can draw its Subgroup Lattice:

DrawSubgroupLattice(GaloisGroup(poly, x), 'indices')

But according to Galois's theory, each subgroup represents an intermediate field. As far as I know, ⑤⑥⑦⑧ are Q(r₁),Q(r₂),Q(r₃) and Q(r₄), respectively, where r_i is the root of equation x^4+8x+12. But I have no idea what fields ②③④⑨ means. How do you calculate out those intermediate fields with maple?

This code can draw the subgroup lattice:

DrawSubgroupLattice(GaloisGroup(x^3 - 2, x), 'indices')

But I really want to know what the extension of field about each subgroup corresponds to, like this:

The root of the polynomial r₁=2^1/3, r₂=2^1/3(-1+sqrt(3) I)/2, r₂=2^1/3(-1-sqrt(3) I)/2 in the above graph. Further, how do we draw the extension relation of this polynomial:

x⁵+15x+44

The maple can draw this graph? If maple can't draw it, what software can?

As we know, choosing different generators can all represent the same group. But how can I get the combination of these different generators by maple? For example, how do I get the following different combinations of generators of C₆?

If I want to get an irreducible polynomial with root sqrt(5) and minimum degree, I can use:

with(PolynomialTools):
MinimalPolynomial(sqrt(5), x)

But what if I want this irreducible polynomial of the minimum degree to have roots in both sqrt(2), sqrt(3), and sqrt(5)? How do I get this polynomial?

ps: The coefficients are all integers