Number Theory

1. Transcendental Number Theory
   1. Definitions and Concepts
      1. Differentiation between algebraic and transcendental numbers
         1. Algebraic numbers: roots of non-zero polynomial equations with rational coefficients
         2. Transcendental numbers: not roots of any non-zero polynomial equation with rational coefficients
      2. Historical background and significance
         1. Discovery and recognition in mathematical history
         2. Impact on analysis and algebra
   2. Key Theorems and Results
      1. Liouville's Theorem
         1. Statement and implications
         2. Liouville numbers: numbers with exceptionally good rational approximations
         3. Examples and proofs highlighting transcendental characteristics
      2. Lindemann–Weierstrass Theorem
         1. Generalization of the transcendence of e and π
         2. Proof outline and significance
         3. Consequences for the impossibility of certain constructions with straightedge and compass (e.g., squaring the circle)
      3. Baker's Theorem
         1. Extension of transcendence results to linear forms in logarithms
         2. Effects on Diophantine approximations and applications
         3. Notable examples and specific case studies
   3. Methods and Techniques
      1. Approximations and Measures
         1. Use of continued fractions for approximating algebraic numbers
         2. Measures of irrationality and transcendence
         3. Quantitative mechanisms for attaining proofs
      2. Applications of Diophantine Approximation
         1. Integral and rational solutions to Diophantine equations
         2. Strategies for bounding errors and achieving precise approximations
   4. Related Advanced Topics
      1. Gelfond-Schneider Theorem
         1. Proof of the transcendence of numbers like a^b when a is algebraic and not 0 or 1, and b is an irrational algebraic number
         2. Historical development and impact on broader mathematical theories
      2. Schanuel's Conjecture
         1. Proposed extension of transcendental results to exponential fields
         2. Current status and attempts at proof
      3. Transcendence in Complex Analysis
         1. Role in analytic functions and differential equations
         2. Relevance to exponential and logarithmic identities
   5. Open Problems and Research Directions
      1. Ongoing conjectures and unresolved questions
         1. Developments and insights from recent research
         2. Implications for fields such as algebraic geometry and mathematical logic
   6. Applications and Influence
      1. Interaction with other branches of mathematics
         1. Connections with algebraic geometry and number fields
         2. Influence on cryptography and computational mathematics
      2. Non-mathematical influences
         1. Philosophical interpretations and the concept of ‘mathematical existence’