Ask Your Question

Subgroup of unit group

asked 2022-02-02 23:32:08 +0100

AlonAmit gravatar image
k = CyclotomicField(5)
U = k.unit_group()
v = U.gens()
u = U.gens_values()
T1 = U.subgroup([v[0]])
T2 = U.subgroup([u[0]])

This code attempts to generate a subgroup of the unit group of $\mathbb{Q}(\zeta_{5})$. The subgroup call for T1 works fine. The one for T2 breaks. This is close to bug #18863 that was already fixed, but this version of the problem persists.

The elements of u are recognized as elements of the group U. Still, Sage is unable to compute the subgroup generated by those elements. For my use case, it's important to manipulate the elements of u as elements in the field (including addition), which I don't think I can do with the elements of v.

edit retag flag offensive close merge delete


Welcome to Ask Sage! Thank you for your question.

slelievre gravatar imageslelievre ( 2022-02-02 23:34:10 +0100 )edit

1 Answer

Sort by ยป oldest newest most voted

answered 2022-02-03 11:01:37 +0100

rburing gravatar image

It would be nice if the conversion (from number field element to element of the "abstract" unit group) was automated. Currently you can just do the conversion into the abstract group manually:

sage: U.subgroup([U(-u[0]^3 + u[1]^2 - u[1] + 1)])
Multiplicative Abelian subgroup isomorphic to C2 x C5 generated by {u0^9}
sage: zeta5 = k.gen()
sage: U.subgroup([U(zeta5^3 + zeta5^2 + zeta5 + 1)])
Multiplicative Abelian subgroup isomorphic to C2 x C5 generated by {u0^9}
edit flag offensive delete link more


I see! That's perfect, thank you.

AlonAmit gravatar imageAlonAmit ( 2022-02-03 17:10:57 +0100 )edit

@rburing quick follow up if that's ok - how can I determine the index of the subgroup inside U, and find coset representatives? I've scanned the documentation and couldn't see suitable methods. Checking individual elements of U for being included in the subgroup doesn't seem to work consistently either. Happy to start a separate question if that's better.

AlonAmit gravatar imageAlonAmit ( 2022-02-03 20:35:01 +0100 )edit

Separate question please.

slelievre gravatar imageslelievre ( 2022-02-04 00:31:45 +0100 )edit

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

1 follower


Asked: 2022-02-02 23:32:08 +0100

Seen: 125 times

Last updated: Feb 03 '22