Number Fields

• Introduction to Number Fields
  • The variable \(x\)
  • Using tab completion to get the methods of an object
  • Symbolic Expressions
  • sqrt(2) in Pari and Magma
  • Numerically evaluating sqrt(2)
  • Arithmetic with sqrt(2)
  • Adjoining a symbolic expression
  • Coercion: QQ[a] versus QQ(a)
  • Solving a cubic equation
  • Viewing complicated symbolic expressions
  • Adjoining a root of the cubic
• Number Fields: Galois Groups and Class Groups
  • Galois Groups
  • Some more Galois groups
  • Magma’s Galois group command
  • Computing complex embeddings
  • Class Numbers and Class Groups
  • Quadratic imaginary fields with class number 1
  • Enumerating quadratic imaginary fields with class number 1
  • Class number 1 fields
  • Class numbers of cyclotomic fields
  • Assuming conjectures to speed computations
  • Class group structure
  • Arithmetic in the class group
• Orders and Relative Extensions
  • Orders in Number Fields
  • Constructing the order with given generators
  • Computing Maximal Orders
  • Functionality for non-maximal orders is minimal
  • Relative Extensions
  • Constructing a relative number field step by step
  • Functions on relative number fields
  • Extra structure on relative number fields
  • Arbitrary towers of relative number fields
  • Relative number field arithmetic can be slow