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.