Generators of a Galois group in SAGEMath

Question

I have a (potentially silly) question about a problem I've been having with SAGEMath.

Let $K$ be the maximal totally real subfield of the cyclotomic field of conductor $24$, i.e. the number field with defining polynomial $x^4-4x^2+1$. I have defined this as such in SAGE:

K.<f>=NumberField(x^4-4*x^2+1)

I then go on to define the Galois group of $K$:

G=K.galois_group()

This group has order $4$. If we check the generators of this group by writing G.gens(), we get

[(1,2)(3,4), (1,4)(2,3)]

If I understand this correctly, this means there are two generators of the group $G$. However, if I write

G.<g,h>=K.galois_group()

I receive an error saying that the number of names needs to match the number of generators of $G$. Writing G.<g>=K.galois_group() works, but g here has order $2$. Why is it that I can't define $G$ in terms of two generators in SAGE?

Answer 1

In Sagemath, the syntax OBJECT.<a,b,c,...> = ... works if the right side of the equation is properly set up for this; among other things, I believe it looks for an argument called names and tries to use a,b,c,... for that argument. In the case of galois_group, one of the arguments is names:

• "names" - a string giving a name for the generator of the Galois closure of self, when this field is not Galois.

So in this case, G.<a,b,c, ...> = K.galois_group() is not trying to define generators of the group, but instead a single generator for the Galois closure of K.

To actually get the group generators, you can instead define G = K.galois_group() and then do g, h = G.gens().

Edit: I hadn't read the newest comments before posting this, and I now see that this is repeating some of the same information given by @durianice. I've converted this answer to "Community wiki."