Prove (or disprove): Define \(T_n(x) \) as the Chebyshev polynomial of the first kind with degree \(n\). If \(p\) is an odd prime, then \(\sqrt{\frac{T_p(x) - 1}{x-1}} \) is an irreducible polynomials over the rational numbers.

By definition \( T_n(x) = \cos(n \cos^{-1}(x)) \). Listing out the first few odd primes of \(n\) shows that (for simplicity sake, we take the positive root only)

\( T_3(x) - 1 = (4x^3-3x) - 1 = (x-1)(2x+1)^2 \\ \Rightarrow \sqrt{\frac{T_3(x) - 1}{x-1}} = 2x + 1 \)

\( T_5(x) - 1 = (16x^5-20x^3+5x) - 1 = (x-1)(4x^2+2x-1)^2 \\ \Rightarrow \sqrt{\frac{T_5(x) - 1}{x-1}} = 4x^2+2x-1 \)

\( T_7(x) - 1 = (64x^7-112x^5+56x^3-7x) - 1 = (x-1)(8x^3+4x^2-4x-1)^2 \\ \Rightarrow \sqrt{\frac{T_7(x) - 1}{x-1}} = 8x^3+4x^2-4x-1 \)

\(\begin{eqnarray} T_{11}(x) - 1 &=& (1024 x^{11}-2816 x^9+2816 x^7-1232 x^5+220 x^3-11x) - 1 \\ &=& (x-1) (32 x^5+16 x^4-32 x^3-12 x^2+6 x+1)^2 \\ \Rightarrow \sqrt{\frac{T_{11}(x) - 1}{x-1}} &=& 32 x^5+16 x^4-32 x^3-12 x^2+6 x+1 \end{eqnarray}\)

All these polynomials are irreducible polynomials over the rationals. Is it true for all odd prime \(p\)?

## Comments

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TopNewestThe solution is as I expected click here.I hope you don't mind that I copied the problem to math overflow.It uses only the multiplicative property of the degrees of number fields and that \(x^{p-1}+x^{p-2}+...+1\), the pth cyclotomic polynomial, is irreducible (can be proven by shifting x to x+1 and using eisenstein). – Bogdan Simeonov · 2 years, 1 month ago

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– Pi Han Goh · 2 years, 1 month ago

Haha, it's a bit of an overkill to use OverFlow! I was about to post this in StackExchange if there's no answer. Thanks!Log in to reply

– Bogdan Simeonov · 2 years, 1 month ago

I posted it on stackexchange, but no one provided a good solution :DLog in to reply

This post needs to be seen by more people so we can actually solve it lol.Also I still cannot understand why Eisenstein works here (what is special about the coeffs) – Bogdan Simeonov · 2 years, 1 month ago

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I think that the identity \(T_n(x)=cos(n.arccosx)\) can give us the roots of \(T_n(x)-1\) – Bogdan Simeonov · 2 years, 1 month ago

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– Pi Han Goh · 2 years, 1 month ago

I'm listening. Please continue on your elaboration.Log in to reply

– Bogdan Simeonov · 2 years, 1 month ago

Well cosx is 1 only when x is 2.pi.something.So that means that \(n.arrcosx\) must be 2.pi.something.So \(x=cos(2.\pi.k/n)\) are the roots of T_n(x)-1.Problem is, I don't know when they double roots and how to prove that T is their minimal polynomial.It is interesting that these roots are the real parts of the roots of unity.Log in to reply

– Bogdan Simeonov · 2 years, 1 month ago

Also the identity \((T_n(x)-1)/(x-1)(T_n(x)+1)/(x+1)=U_{n-1}\) might help, since we know the factorisation of U.Log in to reply

If you shift x to x+1 the irreducibility of these can be proven with Eisenstein's criterion. – Bogdan Simeonov · 2 years, 1 month ago

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– Bogdan Simeonov · 2 years, 1 month ago

By the way do you have a solution or just spotted that they are irreducible for small values?Vecause I can't figure out what identity to use to make that square root in a better form.Eisenstein still works thoughLog in to reply

my solution here. – Pi Han Goh · 2 years, 1 month ago

I don't have a solution, hence the "or disprove" statement. I got inspired by solving via the Challenge Master notes comment onLog in to reply

– Bogdan Simeonov · 2 years, 1 month ago

So it is perfectly possible that a counterexample exists.Well I'm going to read up on Tschebyscheff Polynomials, any reccomendations?Also, I have found a paper that states that (T_p(x)/x/) is irreducible for all primes p.But I don't know how to deal with that -1.Log in to reply

– Pi Han Goh · 2 years, 1 month ago

Recommendation? Haha, no, I read up a lot before posting this, and I'm still pretty stumped. Link please?Log in to reply

this paper is interesting – Bogdan Simeonov · 2 years, 1 month ago

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– Pi Han Goh · 2 years, 1 month ago

Thanks! But how do I prove that it's true for ALL odd primes \(p\)?Log in to reply