<s>[INST] elementary properties of cyclotomic polynomials How can one rewrite $1+x^2+x^4+x^8+\cdots x^{2^n}$ as a product of cyclotomic polynomials? more general how can we express $1+x^p+\cdots+x^{p^n}$, where $p$ is a prime, in term of  product of cyclotomic polynomials?
 [/INST] Let $p(x)=1+x^2+x^4+x^6...+x^{2n}$. Assume that $n>1$, otherwise when $n=1$, $p(x)=x^2+1$, which is the only irreducible polynomial in this case. The case that $n$ is even, $p(x)=(1+x+x^2+x^3+x^n)(1-x+x^2-x^3+x^n)$. When $n$ is odd, the best possible factorization is $p(x)=(x^2+1)(1+x^4+x^8+x^{4k})$ where $2k+1=n$.
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