A solvable nonic polynomial

Continuing from our demonstration that a certain sextic polynomial, which are not in general solvable, has an explicit factorization, we go on to describe how a class of degree 9 polynomials is solvable. Consider a,b,c,d to be rational integers, not all zero, and the nonic polynomial

F(x) = x^9 + a x^8 + b x^7 + c x^6 + d x^5 - (126 - 56 a + 21 b - 6 c + d)x^4

- (84 - 28 a + 7 b - c)x^3 - (36 - 8a + c)x^2 - (9 - a)x - 1.

The claim is that all such polynomials are in fact solvable!

I will reveal the argument a little later, but it’ll be interesting to see what kind of arguments readers can come up with.

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One thought on “A solvable nonic polynomial

  1. Pingback: On binary forms | The Conscious Mathematician

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