A Family of Solvable Quintics and Septics


x = \frac{-\sqrt{2}\,\eta(2\tau)}{\zeta_{48}\,\eta(\tau)}

where \eta is the Dedekind eta function, and \zeta_{48} is the 48th root of unity.  Then for \tau = \frac{1+\sqrt{-d}}{2} for d = {47, 103}, x is a root of the quintics,

x^5-2x^4+2x^3-x^2+1 = 0

x^5-2x^4+3x^3-3x^2+x+1 = 0

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,

x^5-2x^4+2x^3-x^2+1 = nx(x-1)^2

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,

x^7-2x^6+x^5-x^4-5x^2-6x-4 = n(x-1)x^2(x+1)^2

with discriminant,

d = 4^4(4n^3+99n^2+34n+467)^3 .

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.

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