# Number Theory Seminar

Presented by: Abdullah Al-Shaghay

​Title: “Some Classes of Generalized Cyclotomic Polynomials​​”

​Abstract: ​

For a positive integer $n$ the $n^{th}$ cyclotomic polynomial can be written as
$\Phi_{n}(x)=\prod_{\substack{k=1 \\ (k,n)=1}}^{n} \left(x-e^{\tfrac{2\pi ik}{n}}\right) =\prod_{\substack{k=1 \\ k\in(\mathbb{Z}/n\mathbb{Z})^{\times}}}^{n} \left(x-e^{\tfrac{2\pi ik}{n}}\right).$
When $n=p$ is an odd prime, the $n^{th}$ cyclotomic polynomial has the special form
$\Phi_{p}(x)=\sum_{k=0}^{p-1}x^{k}=x^{p-1}+x^{p-2}+\cdots+x+1.$
These two representations of the cyclotomic polynomials highlight the roots of $\Phi_{n}(x)$ and the coefficients of $\Phi_{n}(x)$ respectively. Continuing with the work of Kwon, J. Lee, and K. Lee and Harrington we investigate the generalization of the cyclotomic polynomials in two distinct ways; one affecting the roots of $\Phi_{n}(x)$ and the other affecting the coefficients of $\Phi_{n}(x)$.

In the final chapter of the thesis we discuss congruences for particular binomial sums and use those congruences to prove results concerning two special cases of Jacobi polynomials, the Chebyshev polynomials and the Legendre polynomials.​

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Lectures, Seminars

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