Reuse of data in rnfkummer/computing order of galois elements without it

[Watson Ladd <watsonbladd@gmail.com>]

Dear all,
I have a program which computes many rnfkummer's over the same base
field K. All the rnfkummers are quadratic extensions. This ends up
taking a very, very long time, particularly as many of the extensions
have galois groups which contain elements of orders which are two
large and are thus thrown out, so the time spent computing the
extensions is wasted.

My questions are as follows:

1: Can anything be done to accelerate the repeated rnfkummers?

2: Is there a way to determine the maximal order of an element of the
Galois group of the resulting field without computing the field via
rnfkummer? I know about bnrgaloismat, but it doesn't seem as though
the resulting representation of the galois group can be used to find
the order of elements easily.

An affirmative answer to 2 would render 1 superfluous.

Sincerely,
Watson