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Let $T(n,x)$ be the nth Chebyshev polynomial of the first kind and let $U(n-1,x)$ the $(n-1)$th Chebyshev polynomial of the second kind. Would any one kindly help show that

1) $n$ is prime iff $T(n,x)$ is irreducible in $\mathbb{Z}[x]$.

2) $n$ is prime iff $U(n-1,x)$, expressed in powers of $(x^2-1)$, is irreducible in $\mathbb{Z}[x]$.

Many Thanks!!


No "ordering" is implied here. The wording of the question was done in similar fashion as any question in any math/research question. I do not see how a person who is asking for help would be ordering people to help.

Any way back to the topic,

For the first part of the question, I noticed that if n is prime, then T(n,x) satisfies Eisenstein's Irreducibility Criterion. But I am not sure how to show if T(n,x) is irreducible then n is prime.

  • 2
    $$T_n(x)=\frac12\left(\left(x+\sqrt{x^2-1}\right)^n + \left(x-\sqrt{x^2-1}\right)^n\right)$$ might be helpful.2012-02-14
  • 1
    Perhaps the property that $T_{mn}(x) = T_n(T_m(x))$ might be useful.2012-02-15

3 Answers 3