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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

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