ubuntu2404
\relax
\providecommand\zref@newlabel[2]{}
\providecommand \babel@aux [2]{\global \let \babel@toc \@gobbletwo }
\@nameuse{bbl@beforestart}
\bibstyle{biblatex}
\bibdata{main-blx,references}
\citation{biblatex-control}
\abx@aux@refcontext{nyt/global//global/global/global}
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\babel@aux{english}{}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}\protected@file@percent }
\newlabel{sec:introduction}{{1}{1}{Introduction}{section.1}{}}
\newlabel{sec:introduction@cref}{{[section][1][]1}{[1][1][]1}{}{}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Cyclotomic Polynomials and Basic Properties}{2}{section.2}\protected@file@percent }
\newlabel{sec:cyclotomic}{{2}{2}{Cyclotomic Polynomials and Basic Properties}{section.2}{}}
\newlabel{sec:cyclotomic@cref}{{[section][2][]2}{[1][1][]2}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Computational Construction of Cyclotomic Polynomials}{2}{subsection.2.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Properties and Relationships}{2}{subsection.2.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Visualization of Cyclotomic Polynomial Roots}{3}{subsection.2.3}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Visualization of $n$-th roots of unity on the unit circle for various values of $n$. Blue points represent all $n$-th roots of unity, while red points highlight the primitive roots that are zeros of the cyclotomic polynomial $\Phi _n(x)$. The number of red points equals $\varphi (n)$, confirming the degree formula.}}{4}{figure.caption.1}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:cyclotomic_roots}{{1}{4}{Visualization of $n$-th roots of unity on the unit circle for various values of $n$. Blue points represent all $n$-th roots of unity, while red points highlight the primitive roots that are zeros of the cyclotomic polynomial $\Phi _n(x)$. The number of red points equals $\varphi (n)$, confirming the degree formula}{figure.caption.1}{}}
\newlabel{fig:cyclotomic_roots@cref}{{[figure][1][]1}{[1][3][]4}{}{}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Galois Theory of Cyclotomic Fields}{4}{section.3}\protected@file@percent }
\newlabel{sec:galois}{{3}{4}{Galois Theory of Cyclotomic Fields}{section.3}{}}
\newlabel{sec:galois@cref}{{[section][3][]3}{[1][4][]4}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Computational Analysis of Galois Groups}{4}{subsection.3.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Ramification in Cyclotomic Fields}{4}{subsection.3.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4}Applications and Advanced Topics}{5}{section.4}\protected@file@percent }
\newlabel{sec:applications}{{4}{5}{Applications and Advanced Topics}{section.4}{}}
\newlabel{sec:applications@cref}{{[section][4][]4}{[1][5][]5}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Connection to Quadratic Reciprocity}{5}{subsection.4.1}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Visualization of Legendre symbols $(p/q)$ for odd primes. Red indicates $-1$, white indicates $0$ (which doesn't occur for distinct primes), and blue indicates $+1$. The asymmetry demonstrates the reciprocity law: $(p/q)(q/p) = (-1)^{\frac {p-1}{2} \cdot \frac {q-1}{2}}$.}}{5}{figure.caption.2}\protected@file@percent }
\newlabel{fig:quadratic_reciprocity}{{2}{5}{Visualization of Legendre symbols $(p/q)$ for odd primes. Red indicates $-1$, white indicates $0$ (which doesn't occur for distinct primes), and blue indicates $+1$. The asymmetry demonstrates the reciprocity law: $(p/q)(q/p) = (-1)^{\frac {p-1}{2} \cdot \frac {q-1}{2}}$}{figure.caption.2}{}}
\newlabel{fig:quadratic_reciprocity@cref}{{[figure][2][]2}{[1][5][]5}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Class Numbers and Computational Challenges}{6}{subsection.4.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5}Computational Complexity and Algorithms}{6}{section.5}\protected@file@percent }
\newlabel{sec:complexity}{{5}{6}{Computational Complexity and Algorithms}{section.5}{}}
\newlabel{sec:complexity@cref}{{[section][5][]5}{[1][5][]6}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Factorization Algorithms for Cyclotomic Polynomials}{6}{subsection.5.1}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Computational complexity analysis of cyclotomic polynomials. (Top left) Degree growth following Euler's totient function. (Top right) Coefficient height growth showing exponential behavior for certain values. (Bottom left) Computation time scaling with $n$. (Bottom right) Relationship between computation time and polynomial degree on log-log scale.}}{6}{figure.caption.3}\protected@file@percent }
\newlabel{fig:computational_complexity}{{3}{6}{Computational complexity analysis of cyclotomic polynomials. (Top left) Degree growth following Euler's totient function. (Top right) Coefficient height growth showing exponential behavior for certain values. (Bottom left) Computation time scaling with $n$. (Bottom right) Relationship between computation time and polynomial degree on log-log scale}{figure.caption.3}{}}
\newlabel{fig:computational_complexity@cref}{{[figure][3][]3}{[1][5][]6}{}{}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Conclusions and Future Directions}{6}{section.6}\protected@file@percent }
\newlabel{sec:conclusions}{{6}{6}{Conclusions and Future Directions}{section.6}{}}
\newlabel{sec:conclusions@cref}{{[section][6][]6}{[1][6][]6}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Key Findings}{7}{subsection.6.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Future Research Directions}{7}{subsection.6.2}\protected@file@percent }
\abx@aux@read@bbl@mdfivesum{C509C3014AE4E0F7AB9C902C559D8615}
\gdef \@abspage@last{7}