Number Theory

1. Algebraic Number Theory
   1. Algebraic Numbers and Integers
      1. Definition and Properties
         1. Algebraic numbers as roots of polynomials with rational coefficients
         2. Distinction between algebraic integers and non-integers
         3. Examples and non-examples of algebraic numbers
      2. Minimal Polynomial
         1. Uniqueness and existence of minimal polynomials
         2. Construction of minimal polynomials
         3. Degree of an algebraic number
      3. Field Extensions
         1. Definition and examples of field extensions
         2. Algebraic extensions versus transcendental extensions
         3. Degree of extensions and the Tower Law
   2. Number Fields
      1. Definition and Examples
         1. Definition of number fields as finite extensions of the rational numbers
         2. Example: Quadratic and cyclotomic fields
      2. Rings of Integers
         1. Definitions and comparison with regular integers
         2. Norm and trace of an algebraic integer
         3. Algebraic integers forming a ring
      3. Dedekind Domains
         1. Definition and properties of Dedekind domains
         2. Role in unique factorization of ideals
         3. Relationship with rings of integers in number fields
   3. Ideals and Factorization
      1. Concepts of Ideals
         1. Definition of prime and maximal ideals
         2. Operations with ideals: sum, product, and intersection
      2. Principal Ideal Domains (PIDs)
         1. Definition and characteristics
         2. Examples within rings of integers
         3. Non-uniqueness of factorization in general integral domains
      3. Unique Factorization of Ideals
         1. Fundamental theorem of ideal theory for Dedekind domains
         2. Contrast with unique factorization of elements
         3. Examples illustrating unique factorization of ideals
   4. Galois Theory and Number Fields
      1. Introduction to Galois Theory
         1. Connection between field extensions and permutation groups
         2. Galois groups and their properties
      2. Applications to Number Theory
         1. Solvability of equations by radicals
         2. Analyzing structure and symmetries of number fields
      3. Kronecker–Weber theorem
         1. Cyclotomic fields as extensions of rational numbers
   5. Class Groups and Class Numbers
      1. Definition of Class Groups
         1. Concept of ideal classes and formation of class group
         2. Class number as a measure of the failure of unique factorization
      2. Computation and Examples
         1. Techniques for computing class numbers
         2. Examples in quadratic fields
      3. Significance in the Study of Number Fields
         1. Role in understanding the arithmetic of the field
         2. Relationship with the distribution of prime ideals
   6. Units in Number Fields
      1. Dirichlet's Unit Theorem
         1. Statement and implications of the theorem
         2. Structure of the group of units in a number field
      2. Calculation Techniques
         1. Finding fundamental units
         2. Examples in quadratic number fields
      3. Applications and Further Insights
         1. Analyses of Pell's equation using units
         2. Connections with algebraic K-theory and regulator elements