(*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.