(Under construction)
Archive for the ‘Uncategorized’ Category
9 Apr
A Family of Solvable Quintics and Septics
Define,
where is the Dedekind eta function, and is the 48th root of unity. Then for for d = {47, 103}, x is a root of the quintics,
respectively. Note that the class number h(d) of both is 5. It turns out these belong to a family of solvable quintics found by Kondo and Brumer,
for any n, and where the two examples are n = {0, -1}. A similar one for septics can be deduced from the examples in Kluner’s A Database For Number Fields as,
with discriminant,
.
The case n = 0 implies d = 467 and, perhaps not surprisingly, the class number of h(-467) = 7. However, since 467 does not have form 8m+7, then the eta quotient will be not be an algebraic number of degree h(-d).
To find a solvable family, it’s almost as if all you need is to find one right solvable equation, affix the right n-multiple of a polynomial on the RHS, and the whole family will remain solvable.