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In Heptadecagon page:

The trigonometric functions $\cos(\pi/17)$ and $\cos(2\pi/17)$ are both algebraic numbers of degree 8 given respectively by:

$\cos(\pi/17) = (256x^8-128x^7-448x^6+192x^5+240x^4-80x^3-40x^2+8x+1)_8\\ \cos(2\pi/17) = (256x^8+128x^7-448x^6-192x^5+240x^4+80x^3-40x^2-8x+1)_8$

I don't know what x is and what the subscript $8$ means.

If it means $256x^8-128x^7-448x^6+192x^5+240x^4-80x^3-40x^2+8x+1=0$, how can we prove that $x$ is a constructible number?

(I know how to prove $\cos(\pi/17)$'s constructibility which is found here. But I don't understand the equations mentioned above.)

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auntyellow
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  • It’s a root of that polynomial – J. W. Tanner Jul 08 '21 at 03:38
  • See https://en.wikipedia.org/wiki/Algebraic_number for a definition – Ross Millikan Jul 08 '21 at 03:40
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    I'm not entirely sure of the subscript notation, but it happens to be true that both $\cos(\pi/17)$ and $\cos(2\pi/17)$ are the greatest roots of their respective minimal polynomials, hence in 8th positions when the roots are arranged in order. – Blue Jul 08 '21 at 03:46
  • Meanwhile, the whole business is in Galois Theory by Cox. Chapter 9 gives Gauss periods in modern language. Chapter 10 is about constructible numbers. https://www.google.com/books/edition/Galois_Theory/vBKrOch1AkYC?hl=en&gbpv=1&printsec=frontcover – Will Jagy Jul 08 '21 at 03:48
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    If you want to see that polynomial equation derived, then it follows in a straightforward manner starting from the fact $$2\cos(2\pi/17)=\zeta+\zeta^{-1},$$ where $\zeta=e^{2\pi i/17}$ is the prescribed complex root of unity, together with the fact $$\zeta^8+\zeta^7+\cdots+\zeta+1+\zeta^{-1}+\cdots+\zeta^{-7}+\zeta^{-8}=0$$ that follows from the formula for a geometric sum. This old answer of mine describes the technique and derives a polynomial equation for the simpler $2\cos(2\pi/5)$ as an example. – Jyrki Lahtonen Jul 08 '21 at 05:11
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    A description of how constructibility follows from using $\zeta+\zeta^{-1}$ can be found from many textbooks. Such as the one Will Jagy linked to. – Jyrki Lahtonen Jul 08 '21 at 05:25

1 Answers1

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The subscript $8$ is a notation that is used to identify a particular root of the enclosed polynomial. This is not a commonly used notation, but the ordering is (poorly) documented in Mathematica:

The ordering used by Root[f,k] takes real roots to come before complex ones, and takes complex conjugate pairs of roots to be adjacent.

In fact, the ordering is in ascending magnitude of the real-valued roots, followed by ordering the complex-valued roots in a way that is not obvious to me; all I can say is that they are arranged in increasing order of real part, but there does not seem to be a consistent rule for the ordering of roots with the same real part but different imaginary parts. This is more of a question for Mathematica.SE as it is specific to the implementation in Mathematica. But suffice it to say this is not a standard notation.

In any case, your specific situation is not ambiguous because the roots of the minimal polynomials for $\cos \frac{2\pi}{17}$ and $\cos \frac{\pi}{17}$ are all real-valued: there are no complex-valued roots. So when the subscript $8$ is used, it is referring to the unique largest real root.

As for $x$, this is just a placeholder variable. It makes no difference if it is $x$ or $y$ or "ducks." The point is that there is a polynomial $f$ of degree $8$ of some variable, and the solution of the equation $f = 0$ for that variable leads to $8$ real-valued roots, the largest of which is $\cos \frac{2\pi}{17}$ or $\cos \frac{\pi}{17}$ depending on which polynomial we are talking about.

But you are correct: just by looking at the polynomial, we cannot immediately tell whether such a polynomial has roots that are constructible--i.e., they are expressible using only a finite number of additions, subtractions, multiplications, divisions and square root operations on integers.

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