The Radical Solution of a Solvable 17th Degree Equation
Tito Piezas III
(Note: Instead of using superscripts as I normally do, in this short article I’ve used the hat symbol ^ so the equations can easily be copied and pasted onto Mathematica or Maple for those who wish to see the 16th-deg resolvent themselves.)
Here's a “natural” solvable 17th-deg eqn with small coefficients:
x^{17}-6 x^{16}-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 x^{11}-x^{10}-4 x^9-11 x^8-7 x^7-13 x^6-x^5+x^3+x^2+x-1 = 0 (eq.1)
Its unique real root is exactly given by (in Mathematica) as x = zeta_48 DedekindEta[tau]/(Sqrt[2]DedekindEta[2tau]) = 9.1630942..., with the root of unity zeta_48 = exp(2Pi I/48), tau = (1+Sqrt[-d])/2, and d = 383. This d has class number h(-d) = 17.
To solve this, depress eq.1 (get rid of its xn-1 term), by letting x = (y+6)/17 to get,
y^{17}-11832 y^{15}-1124346 y^{14}-55393735 y^{13}-1784741617 y^{12}-41171464807 y^{11}-711423456455 y^{10}-9455898295636 y^9-99724287747103 y^8-887992943070295 y^7-7665207188897171 y^6-70479807472769473 y^5-592167373130143650 y^4-3496187093606980919 y^3-8695712981307573757 y^2+68265051092799270505 y-427806967360317821039 = 0 (eq.2)
Its 16th-deg resolvent, a polynomial with INTEGER coefficients, call this R_16, has roots,
z_k = ((y1 + w^{k}y2 + w^{2k}y8 + w^{3k}y7 + w^{4k}y16 + w^{5k}y4 + w^{6k}y12 + w^{7k}y15 + w^{8k}y11 + w^{9k}y10 + w^{10k}y14 + w^{11k}y13 + w^{12k}y5 + w^{13k}y17 + w^{14k}y6 + w^{15k}y9 + w^{16k}y3)/17)^17
for k = {1 to 16}, where w is any complex 17th root of unity. This particular R_16 has all z_k as real numbers.
Note the specific arrangement of the y_n. There are 16! ≈ 2 x 10^13 possible permutations of the y_n, and out of that huge number, there are only 16 such that R_16 has integer coefficients, and we have given one of them. Of course, I used a short cut to find it, because even if your computer can check a million permutations a second, it would still take about 8 months to go through them all. The short cut took less than two hours to find R_16. The y_n follows the root object Root[poly, n] ordering in Mathematica. Approximately, these are,
y1 = 149.7726
{y2, y3} = -27.62 -/+ 18.49 i
{y4, y5} = -21.61 -/+ 7.52 i
{y6, y7} = -16.58 -/+ 6.34 i
{y8, y9} = -10.57 -/+ 15.32 i
{y10, y11} = -5.02 -/+ 13.71 i
{y12, y13} = -2.34 -/+ 13.15 i
{y14, y15} = 2.57 -/+ 2.60 i
{y16, y17} = 6.31 -/+ 7.04 i
R_16 has extremely large integer coefficients, with the largest being the 248-digit constant term 429534618434587^17 which, naturally enough, is a 17th power. (Note: R_16 can easily be formed using 500-digit precision or more on the y_n, and multiplying the 16 factors together to form the polynomial. However, since it involves extremely large integers of 200+ digits, it is understandable if I do not explicitly and tediously type them down here. But if you have Mathematica or Maple, you can easily re-construct R_16 using the details described in this article.)
The polynomial R_16 can then be factored into two octics over the extension Sqrt[17]. This, in turn, can be factored into 2 quartics over Sqrt[2(17+Sqrt[17])]. This can be factored further into 2 quadratics using an expression involved in the 17th root of unity. Apparently, in general, to solve the 16-deg resolvent of an irreducible but solvable 17-deg equation, only square roots of square roots of square roots, etc, are needed.
The real root of eq.2 in radicals is then,
y1 = z1^(1/17) + z2^(1/17) + z3^(1/17) + … + z16^(1/17) = 149.7726…
The next step, of course, is degree p = 19. Unfortunately, to find the 18th-deg resolvent R_18 of an irreducible but solvable 19th-deg equation with rational coefficients, there is a small matter of 18! ≈ 6.4 x 10^15 possible permutations of the y_n, only 18 of which will be such that R_18 is a polynomial with rational coefficients…
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© Tito Piezas III, Sept 2011
You can email author at tpiezas@gmail.com.