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A Conversational Introduction to Algebraic Number Theory

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Table of Contents

  • Getting our feet wet
  • Cast of characters
  • Quadratic number fields: First steps
  • Paradise lost--and found
  • Euclidean quadratic fields
  • Ideal theory for quadratic fields
  • Prime ideals in quadratic number rings
  • Units in quadratic number rings
  • A touch of class
  • Measuring the failure of unique factorization
  • Euler's prime-producing polynomial and the criterion of Frobenius-Rabinowitsch
  • Interlude: Lattice points
  • Back to basics: Starting over with arbitrary number fields
  • Integral bases: From theory to practice, and back
  • Ideal theory in general number rings
  • Finiteness of the class group and the arithmetic of $overline{mathbb{Z}}$
  • Prime decomposition in general number rings
  • Dirichlet's units theorem, I
  • A case study: Units in $mathbb{Z}[sqrt[3]{2}]$ and the Diophantine equation $X^3-2Y^3=pm1$
  • Dirichlet's units theorem, II
  • More Minkowski magic, with a cameo appearance by Hermite
  • Dedekind's discriminant theorem
  • The quadratic Gauss sum
  • Ideal density in quadratic number fields
  • Dirichlet's class number formula
  • Three miraculous appearances of quadratic class numbers
  • Index.

About the Author

Paul Pollack, University of Georgia, Athens, GA.

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