I was intending to extend the function K.selmer_group(S,m) so that it
could used for K=QQ without having treat this as a special case in my
code, or use a trick such as defining Q=NumberField(x-1):
sage: Q.<t>=NumberField(x-1)
sage: Q.selmer_group(Q.primes_above(7),2)
[7, -1]
sage: Q.selmer_group(Q.primes_above(5),2)
[5, -1]
This output agrees with my definition of the K-selmer-group usually
denoted K(S,m), as the subgroup of K^*/(K^*)^m whose representative
elements a in K^* satisfy m | ord_p(a) for all p not in S. But the
docstring for the selmer_group function claims something slightly
different:
Compute the Selmer group `K(S,m)`. This is the subgroup of
`K^\times/(K^\times)^m` consisting of elements `a` such that
`K(\sqrt[m]{a})/K` is unramified at all primes of `K` outside
of `S`.
(The function returns generators of the group rather than the group
itself, but I don't mind that for now and tha is not what I am talking
about here.)
According to this, QQ([5],2) should be [5] and QQ([7],2) should be
[-7], I think. And in general for m=2, the output should have a
generator +p for primes p in S which are 1 (mod 4) and -p when p=3
(mod 4), and both -1 and +2 when 2 is in S.
I suggest that, rather than change the function to do what it says it
does (which would be rather delicate for a general number field and
general m), that the docstring is changed to something like "my"
definition above. Do people agree? Then the above output is correct
and I can continue to implement the version for QQ itself.
John
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