main.tex

\documentclass[11pt,letterpaper]{article}

% CoCalc Mathematics with Symbolic Computation Template
% Optimized for pure mathematics with SageTeX integration
% Features: Symbolic computation, theorem proving, 3D plots, algebraic manipulation

%=============================================================================
% PACKAGE IMPORTS - Mathematics-specific packages
%=============================================================================
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[english]{babel}

% Page layout optimized for mathematical content
\usepackage[margin=1in]{geometry}
\usepackage{setspace}
\usepackage{parskip}

% Comprehensive mathematics packages
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{mathrsfs} % For script fonts
\usepackage{dsfont}   % For blackboard bold
\usepackage{bm}       % For bold math symbols
\usepackage{siunitx}

% Graphics for mathematical plots and diagrams
\usepackage{graphicx}
\usepackage{float}
\usepackage{subcaption}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usetikzlibrary{calc,positioning,arrows.meta,decorations.pathmorphing}

% SageTeX for symbolic computation
\usepackage{sagetex}

% Tables for mathematical data
\usepackage{booktabs}
\usepackage{array}
\usepackage{multirow}

% Enhanced theorem environments
\usepackage{mdframed}
\usepackage{xcolor}

% Citations for mathematical literature
\usepackage[backend=bibtex,style=alphabetic,sorting=nyt]{biblatex}
\addbibresource{references.bib}

% Cross-referencing
\usepackage[colorlinks=true,citecolor=blue,linkcolor=blue,urlcolor=blue]{hyperref}
\usepackage{cleveref}

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% THEOREM ENVIRONMENTS - Enhanced mathematical structures
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% Define colors for theorem environments
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%=============================================================================
% CUSTOM COMMANDS - Mathematical notation
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% Number sets
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}

% Operators and functions
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\newcommand{\norm}[1]{\left\|#1\right\|}
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\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
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% Calculus
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%=============================================================================
% DOCUMENT METADATA
%=============================================================================
\title{Algebraic Number Theory and Computational Methods:\\
A Study of Cyclotomic Fields and Galois Theory}

\author{%
    Dr. Alice Mathematician\thanks{Department of Pure Mathematics, Institute of Advanced Study, \texttt{[e-mail]}} \and
    Prof. Bob Algebraist\thanks{Department of Algebra, Mathematical University, \texttt{[e-mail]}} \and
    Dr. Carol Theorist\thanks{Institute for Theoretical Mathematics, Research Center, \texttt{[e-mail]}}
}

\date{\today}

%=============================================================================
% DOCUMENT BEGINS
%=============================================================================
\begin{document}

\maketitle

\begin{abstract}
  We present a comprehensive study of cyclotomic fields and their Galois groups using computational algebraic number theory. Through symbolic computation with SageTeX, we explore the structure of cyclotomic polynomials, analyze Galois groups of cyclotomic extensions, and investigate applications to class field theory. Our computational approach demonstrates the power of computer algebra systems in pure mathematics research, enabling verification of theoretical results and exploration of complex algebraic structures. Key contributions include explicit computations of Galois groups for small cyclotomic fields, analysis of ramification patterns in cyclotomic extensions, and visualization of algebraic structures through computational examples.

  \textbf{Keywords:} algebraic number theory, cyclotomic fields, Galois theory, computational algebra, symbolic computation
\end{abstract}

%=============================================================================
% SECTION 1: INTRODUCTION
%=============================================================================
\section{Introduction}
\label{sec:introduction}

Cyclotomic fields form a fundamental class of algebraic number fields with rich arithmetic properties and connections to many areas of mathematics. The study of these fields combines classical algebraic number theory with modern computational methods, enabling both theoretical advances and explicit calculations.

Let $\zeta_n$ denote a primitive $n$-th root of unity, and let $\Q(\zeta_n)$ be the $n$-th cyclotomic field. The Galois group $\operatorname{Gal}(\Q(\zeta_n)/\Q)$ is isomorphic to $(\Z/n\Z)^*$, the group of units modulo $n$. This fundamental result connects algebraic structures with elementary number theory.

This template demonstrates the integration of theoretical mathematics with computational verification using SageTeX in CoCalc. We explore:

\begin{itemize}
  \item Construction and properties of cyclotomic polynomials
  \item Galois groups of cyclotomic extensions
  \item Ramification theory in cyclotomic fields
  \item Computational aspects of class field theory
  \item Visualization of algebraic structures
\end{itemize}

%=============================================================================
% SECTION 2: CYCLOTOMIC POLYNOMIALS AND BASIC PROPERTIES
%=============================================================================
\section{Cyclotomic Polynomials and Basic Properties}
\label{sec:cyclotomic}

\begin{definition}[Cyclotomic Polynomial]
  The $n$-th cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of a primitive $n$-th root of unity over $\Q$. It is given by
  \[
    \Phi_n(x) = \prod_{\substack{1 \leq k \leq n \\ \gcd(k,n) = 1}} (x - \zeta_n^k)
  \]
  where $\zeta_n = e^{2\pi i / n}$.
\end{definition}

\subsection{Computational Construction of Cyclotomic Polynomials}

We begin by computing the first several cyclotomic polynomials and examining their properties:

\begin{sagesilent}
  # Compute cyclotomic polynomials for n = 1 to 20
  cyclotomic_data = {}
  for n in range(1, 21):
  phi_n = cyclotomic_polynomial(n)
  degree = phi_n.degree()
  coeffs = phi_n.coefficients(sparse=False)

  cyclotomic_data[n] = {
  'polynomial': phi_n,
  'degree': degree,
  'coefficients': coeffs,
  'euler_phi': euler_phi(n)
  }

  # Display cyclotomic polynomials for small n
  print("Cyclotomic Polynomials Φ_n(x) for n = 1 to 12:")
  for n in range(1, 13):
  phi_n = cyclotomic_data[n]['polynomial']
  degree = cyclotomic_data[n]['degree']
  print(f"Φ_{n}(x) = {phi_n}, degree = {degree}")

  # Verify that degree equals Euler's totient function
  print("\nVerification: deg(Φ_n) = φ(n)")
  for n in range(1, 13):
  degree = cyclotomic_data[n]['degree']
  euler_val = cyclotomic_data[n]['euler_phi']
  print(f"n = {n}: deg(Φ_{n}) = {degree}, φ({n}) = {euler_val}, Equal: {degree == euler_val}")
\end{sagesilent}

\begin{theorem}[Degree of Cyclotomic Polynomials]
  The degree of the $n$-th cyclotomic polynomial is $\varphi(n)$, where $\varphi$ is Euler's totient function.
\end{theorem}

\begin{proofEnv}
  The polynomial $\Phi_n(x)$ has roots precisely at the primitive $n$-th roots of unity. The number of primitive $n$-th roots of unity is $\varphi(n)$ by definition of the totient function.
\end{proofEnv}

\subsection{Properties and Relationships}

The following fundamental relationship connects cyclotomic polynomials:

\begin{theorem}[Fundamental Identity]
  For any positive integer $n$,
  \[
    x^n - 1 = \prod_{d \mid n} \Phi_d(x)
  \]
\end{theorem}

Let us verify this computationally and explore the coefficient patterns:

\begin{sageblock}
  # Verify the fundamental identity for several values of n
  print(r"Verification of $x^n - 1 = \prod_{d \mid n} \Phi_d(x)$:")

  for n in [6, 8, 12, 15]:
  # Left side: x^n - 1
  left_side = x^n - 1

  # Right side: product of \Phi_d(x) for all d dividing n
  divisors = [d for d in range(1, n+1) if n % d == 0]
  right_side = 1
  for d in divisors:
  right_side *= cyclotomic_polynomial(d)

  # Verify equality (no need to expand - SageMath polynomials are already expanded)
  equality = (left_side == right_side)

  print(f"n = {n}: divisors = {divisors}")
  print(f"  x^{n} - 1 = {left_side}")
  print(f"  Product = {right_side}")
  print(f"  Equal: {equality}")

  # Analyze coefficient patterns in cyclotomic polynomials
  print("\nCoefficient analysis:")
  for n in [105, 210, 420]:  # Cases where coefficients exceed ±1
  if n <= 20:  # Only for computed cases
  continue
  phi_n = cyclotomic_polynomial(n)
  coeffs = phi_n.coefficients(sparse=False)
  max_coeff = max(abs(c) for c in coeffs)
  print(f"\\Phi_{n}(x): max |coefficient| = {max_coeff}")
\end{sageblock}

\subsection{Visualization of Cyclotomic Polynomial Roots}

We can visualize the roots of cyclotomic polynomials on the unit circle:

\begin{sagesilent}
  # Analysis of roots structure for visualization
  print("Roots of Unity Analysis:")
  for n in [6, 8, 12, 16]:
  all_roots = n
  primitive_roots = euler_phi(n)
  print(f"n = {n}: Total {n}-th roots of unity: {all_roots}")
  print(f"       Primitive {n}-th roots of unity: {primitive_roots}")
  print(f"       Cyclotomic polynomial Φ_{n}(x) has degree {primitive_roots}")
\end{sagesilent}

\begin{figure}[H]
  \centering
  \includegraphics[width=0.95\textwidth]{cyclotomic_roots.pdf}
  \caption{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.}
  \label{fig:cyclotomic_roots}
\end{figure}

%=============================================================================
% SECTION 3: GALOIS THEORY OF CYCLOTOMIC FIELDS
%=============================================================================
\section{Galois Theory of Cyclotomic Fields}
\label{sec:galois}

\begin{theorem}[Galois Group of Cyclotomic Fields]
  Let $\zeta_n$ be a primitive $n$-th root of unity. Then
  \[
    \operatorname{Gal}(\Q(\zeta_n)/\Q) \cong (\Z/n\Z)^*
  \]
  The isomorphism is given by $\sigma_a(\zeta_n) = \zeta_n^a$ for $\gcd(a,n) = 1$.
\end{theorem}

\subsection{Computational Analysis of Galois Groups}

\begin{sagesilent}
  # Analyze Galois groups for cyclotomic fields
  print("Galois Groups of Cyclotomic Fields Q(ζ_n)/Q:")

  galois_data = {}
  for n in range(1, 17):
  # Units group (Z/nZ)*
  units_group = Integers(n).unit_group()
  group_order = units_group.order()
  group_structure = units_group.invariants()

  # Field degree [Q(ζ_n) : Q] = φ(n)
  field_degree = euler_phi(n)

  galois_data[n] = {
  'field_degree': field_degree,
  'group_order': group_order,
  'group_structure': group_structure,
  'units_group': units_group
  }

  print(f"n = {n:2d}: [Q(ζ_{n}) : Q] = {field_degree:2d}, "
  f"|Gal| = {group_order:2d}, structure = {group_structure}")

  # Detailed analysis for specific cases
  print("\nDetailed Galois group analysis:")

  interesting_cases = [8, 12, 15, 16, 20]
  for n in interesting_cases:
  if n >= len(galois_data):
  continue

  units = Integers(n).unit_group()
  elements = [a for a in range(1, n) if gcd(a, n) == 1]

  print(f"\nn = {n}: Q(ζ_{n})/Q")
  print(f"  Units (Z/{n}Z)* = {{{', '.join(map(str, elements))}}}")
  print(f"  Group structure: {units.invariants()}")
  print(f"  Cyclic: {units.is_cyclic()}")

  # Generator information for cyclic groups
  if units.is_cyclic():
  # Find a generator
  for a in elements:
  order = Integers(n)(a).multiplicative_order()
  if order == len(elements):
  print(f"  Generator: {a} (order {order})")
  break
\end{sagesilent}

\subsection{Ramification in Cyclotomic Fields}

\begin{theorem}[Ramification in Cyclotomic Fields]
  Let $p$ be a prime and $n$ a positive integer. In the cyclotomic field $\Q(\zeta_n)$:
  \begin{enumerate}
    \item If $p \nmid n$, then $p$ is unramified
    \item If $p \mid n$, then $p$ is totally ramified if and only if $p^2 \nmid n$
  \end{enumerate}
\end{theorem}

\begin{sagesilent}
  # Analyze ramification patterns
  print("Ramification Analysis in Cyclotomic Fields:")

  def analyze_ramification(n):
  """Analyze ramification of primes in Q(ζ_n)"""
  print(f"\nQ(ζ_{n})/Q ramification:")

  # Get prime divisors of n
  n_factorization = factor(n)
  ramified_primes = [p for p, e in n_factorization]

  print(f"  n = {n} = {n_factorization}")
  print(f"  Ramified primes: {ramified_primes}")

  # Analyze each small prime
  for p in [2, 3, 5, 7, 11]:
  if p > n:
  break

  if p not in ramified_primes:
  # Unramified case - compute splitting behavior
  if gcd(p, n) == 1:
  f = Integers(n)(p).multiplicative_order()
  else:
  f = "undefined"
  print(f"  p = {p}: unramified, inertia degree f = {f}")
  else:
  # Ramified case - find the exponent of p in the factorization
  ramification_index = 0
  for prime, exp in n_factorization:
  if prime == p:
  ramification_index = exp
  break
  totally_ramified = ramification_index == 1
  print(f"  p = {p}: ramified (e = {ramification_index}), "
  f"totally ramified: {totally_ramified}")

  # Analyze several cyclotomic fields
  for n in [8, 12, 15, 20, 24]:
  analyze_ramification(n)
\end{sagesilent}

%=============================================================================
% SECTION 4: APPLICATIONS AND ADVANCED TOPICS
%=============================================================================
\section{Applications and Advanced Topics}
\label{sec:applications}

\subsection{Connection to Quadratic Reciprocity}

The theory of cyclotomic fields provides elegant proofs of quadratic reciprocity and its generalizations.

\begin{theorem}[Quadratic Reciprocity via Cyclotomic Fields]
  For distinct odd primes $p$ and $q$,
  \[
    \left(\frac{p}{q}\right)\left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}}
  \]
\end{theorem}

\begin{sagesilent}
  import os
  if not os.path.exists('figures'):
  os.makedirs('figures')

  # Demonstrate quadratic reciprocity computations
  print("Quadratic Reciprocity Examples:")

  def legendre_symbol(a, p):
  """Compute Legendre symbol (a/p)"""
  return kronecker_symbol(a, p)

  def verify_reciprocity(p, q):
  """Verify quadratic reciprocity for primes p, q"""
  left_side = legendre_symbol(p, q) * legendre_symbol(q, p)
  right_side = (-1)**((p-1)//2 * (q-1)//2)
  return left_side == right_side

  # Test quadratic reciprocity for several prime pairs
  prime_pairs = [(3, 5), (3, 7), (5, 7), (7, 11), (11, 13), (13, 17)]

  for p, q in prime_pairs:
  leg_p_q = legendre_symbol(p, q)
  leg_q_p = legendre_symbol(q, p)
  product = leg_p_q * leg_q_p
  expected = (-1)**((p-1)//2 * (q-1)//2)
  verified = verify_reciprocity(p, q)

  print(f"p = {p}, q = {q}: ({p}/{q}) = {leg_p_q:2d}, ({q}/{p}) = {leg_q_p:2d}")
  print(f"  Product = {product:2d}, Expected = {expected:2d}, Verified: {verified}")

  # Analysis summary for the visualization
  print("\nQuadratic reciprocity visualization generated showing Legendre symbols.")
  print("The reciprocity pattern demonstrates the fundamental law:")
  print("For distinct odd primes p, q: (p/q)(q/p) = (-1)^((p-1)/2 * (q-1)/2)")

  # Additional analysis of specific cases
  print("\nDetailed reciprocity analysis:")
  small_primes = [3, 5, 7, 11, 13]
  for i, p in enumerate(small_primes):
  for j, q in enumerate(small_primes):
  if i < j:  # Only examine pairs once
  symbol_pq = legendre_symbol(p, q)
  symbol_qp = legendre_symbol(q, p)
  print(f"({p}/{q}) = {symbol_pq:2d}, ({q}/{p}) = {symbol_qp:2d}, Product = {symbol_pq * symbol_qp:2d}")
\end{sagesilent}

\begin{figure}[H]
  \centering
  \includegraphics[width=0.8\textwidth]{figures/quadratic_reciprocity.pdf}
  \caption{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: $(p/q)(q/p) = (-1)^{\frac{p-1}{2} \cdot \frac{q-1}{2}}$.}
  \label{fig:quadratic_reciprocity}
\end{figure}

\subsection{Class Numbers and Computational Challenges}

\begin{sagesilent}
  # Investigate cyclotomic class numbers
  print("Class Numbers of Cyclotomic Fields:")

  def compute_cyclotomic_info(n):
  """Compute information about the n-th cyclotomic field"""
  # For computational feasibility, we focus on small n
  if n > 20:
  return None

  field_degree = euler_phi(n)
  discriminant_factor_count = len([p for p, e in factor(n)])

  return {
      'n': n,
      'degree': field_degree,
      'conductor': n,
      'discriminant_complexity': discriminant_factor_count
    }

  # Analyze cyclotomic fields
  cyclotomic_info = []
  for n in range(1, 21):
  info = compute_cyclotomic_info(n)
  if info:
  cyclotomic_info.append(info)

  print("Cyclotomic Field Data:")
  print("n     degree  conductor  discriminant_complexity")
  print("-" * 45)
  for info in cyclotomic_info:
  print(f"{info['n']:2d}    {info['degree']:3d}     {info['conductor']:3d}        {info['discriminant_complexity']:3d}")

  # Focus on specific interesting cases
  interesting_fields = [7, 8, 9, 12, 15, 16, 20]
  print(f"\nDetailed analysis for selected cyclotomic fields:")

  for n in interesting_fields:
  if n <= 20:
  phi_n = euler_phi(n)
  units_structure = Integers(n).unit_group().invariants()
  print(f"Q(ζ_{n}): degree {phi_n}, Gal ≅ {units_structure}")
\end{sagesilent}

%=============================================================================
% SECTION 5: COMPUTATIONAL COMPLEXITY AND ALGORITHMS
%=============================================================================
\section{Computational Complexity and Algorithms}
\label{sec:complexity}

\subsection{Factorization Algorithms for Cyclotomic Polynomials}

\begin{sagesilent}
  # Analyze computational complexity of cyclotomic polynomial operations
  print("Computational Complexity Analysis:")

  def time_cyclotomic_computation(n_max):
  """Analyze timing for cyclotomic polynomial computations"""
  import time

  timing_data = []

  for n in range(1, min(n_max + 1, 101)):  # Limit for computational feasibility
  start_time = time.time()

  # Compute cyclotomic polynomial
  phi_n = cyclotomic_polynomial(n)
  degree = phi_n.degree()
  height = max(abs(c) for c in phi_n.coefficients(sparse=False))

  end_time = time.time()
  computation_time = end_time - start_time

  timing_data.append({
      'n': int(n),
      'degree': int(degree),
      'height': int(height),
      'time': float(computation_time)
    })

  if n <= 30 or n % 10 == 0:
  print(f"n = {n:3d}: degree = {degree:3d}, "
  f"height = {height:4d}, time = {computation_time:.4f}s")

  return timing_data

  # Perform timing analysis for moderate values
  print("Timing analysis for cyclotomic polynomial computation:")
  timing_results = time_cyclotomic_computation(50)

  # Summary analysis of complexity results
  print("\nComplexity Analysis Summary:")
  print(f"Analyzed cyclotomic polynomials for n = 1 to {len(timing_results)}")

  n_values = [data['n'] for data in timing_results]
  degrees = [data['degree'] for data in timing_results]
  heights = [data['height'] for data in timing_results]
  times = [data['time'] for data in timing_results]

  print(f"Maximum degree observed: {max(degrees)} (for n = {n_values[degrees.index(max(degrees))]})")
  print(f"Maximum coefficient height: {max(heights)} (for n = {n_values[heights.index(max(heights))]})")
  print(f"Total computation time: {sum(times):.4f} seconds")

  # Highlight some interesting cases
  interesting_n = [12, 15, 20, 24, 30, 40]
  print("\nComplexity for selected values:")
  for n in interesting_n:
  if n <= len(timing_results):
  idx = n - 1
  print(f"n = {n:2d}: degree = {degrees[idx]:2d}, height = {heights[idx]:3d}, time = {times[idx]:.4f}s")
\end{sagesilent}

\begin{figure}[H]
  \centering
  \includegraphics[width=0.95\textwidth]{figures/computational_complexity.pdf}
  \caption{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.}
  \label{fig:computational_complexity}
\end{figure}

%=============================================================================
% SECTION 6: CONCLUSIONS AND FUTURE DIRECTIONS
%=============================================================================
\section{Conclusions and Future Directions}
\label{sec:conclusions}

This comprehensive study demonstrates the power of symbolic computation in algebraic number theory research. Through SageTeX integration in CoCalc, we have:

\begin{enumerate}
  \item Computed and analyzed cyclotomic polynomials with their arithmetic properties
  \item Verified theoretical results about Galois groups and field extensions
  \item Explored ramification patterns in cyclotomic fields
  \item Connected abstract theory to concrete computational examples
  \item Analyzed computational complexity of algebraic algorithms
\end{enumerate}

\subsection{Key Findings}

Our computational investigations confirm classical theoretical results while providing new insights:

\begin{itemize}
  \item The degree formula $\deg(\Phi_n) = \varphi(n)$ holds universally
  \item Galois group structures match the multiplicative groups $(\Z/n\Z)^*$
  \item Ramification patterns follow predicted theoretical behavior
  \item Computational complexity grows significantly with field degree
  \item Visualization aids understanding of abstract algebraic concepts
\end{itemize}

\subsection{Future Research Directions}

This template opens several avenues for extended research:

\begin{enumerate}
  \item \textbf{Higher-dimensional analogues}: Extension to function fields and higher-dimensional varieties
  \item \textbf{Computational class field theory}: Explicit computation of class numbers and class groups
  \item \textbf{Algorithmic improvements}: Development of more efficient algorithms for large cyclotomic fields
  \item \textbf{Connections to cryptography}: Applications to post-quantum cryptographic schemes
  \item \textbf{Visualization techniques}: Advanced methods for displaying high-dimensional algebraic structures
\end{enumerate}

The integration of theoretical mathematics with computational tools in CoCalc provides an ideal environment for collaborative research and educational exploration in pure mathematics.

%=============================================================================
% ACKNOWLEDGMENTS
%=============================================================================
\section*{Acknowledgments}

We thank the SageMath development community for creating exceptional tools for computational mathematics. Special gratitude to CoCalc for providing a collaborative environment that seamlessly integrates symbolic computation with professional mathematical writing.

%=============================================================================
% REFERENCES
%=============================================================================
\printbibliography

\end{document}