## 125 Reputation

1 years, 270 days

## How to compute the Galois group of reduc...

Maple

If a polynomial is irreducible, you can use GroupTheory:-GaloisGroup or galois to compute its polynomial, for example:

`G := GroupTheory:-GaloisGroup(x^5 - x + 1, x)`

But what if polynomials are reducible? Such as x^5-x+15. How to compute its Galois group by maple? There are some examples to check at the end of this article

## How do you get all the other roots from ...

Maple

Suppose I have a polynomial:

`poly := x^6 - 3*x + 3`

And I know one of its roots and its galois group:

```firstroot := [solve](poly, x)[1]:
G := GaloisGroup(poly, x):```

Since G act transitively on the all roots. How do I get all the other roots of this polynomial just by the firstroot and G, but not by solving the equation.

## How to export the result into SVG file...

Maple 2022

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)`

## How to calculate the all Intermediate fi...

Maple 2022

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 ri 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?

## How to draw the relation of the extensio...

Maple 2022

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=21/3, r2=21/3(-1+sqrt(3) I)/2, r2=21/3(-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?

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