I'm investigating relationships between sets of fundamental units of
complex extensions of the rationals. All the information from here
down to the horizontal line is context, and you can probably
understand my problem without understanding it.
I've constructed a number field L=Q(b) where b is solution to minimal
polynomial of x^6+3 like so:
L.<b>=NumberField(x^6+3)
then I extend this to M = L(c) where c is a solution of t^3 +(b^5-b^3)
*t^2 + (-b^4+2*b^2)*t-b in the following way:
R.<t>=L[]
M.<c>=L.extension(t^3 +(b^5-b^3)*t^2 + (-b^4+2*b^2)*t-b)
________________________________________________________
what all this amounts to is that I have two rather ugly values: b, the
6th root of -3, and c, an ugly solution to a polynomial in b.
The units of M can be attained by the command:
r=M.units()
where r contains 8 values. However, I also have a different set of 8
values from a previous calculation, s, all of whose contents are units
as well. Therefore, for any element r[i] of r there exists an 8-valued
set a such that
r[i]=s[0]^a[0]*s[1]^a[1]*...*s[7]^a[7]
and I am trying to find a, because the previous calculation which took
me from r to s has been mislaid.
My thinking was that if I construct logarithms of both sides,
log(r[i])= a[0]*log(s[0]) + ... +a{7]*log(s[7])
for all i=0,...,7, then I might be able to pull out a matrix of the
different a.
However, I can't figure out a way to do this. Sage doesn't like me
taking logarithms at this point, so I need to embed into RR or CC,
which screams minkowski embedding, but I can't get it to work, because
I don't really know what I'm doing.
Any ideas?
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