Class Number Parity

This book deals with classical questions of Algebraic Number Theory concerning the interplay between units, ideal class groups, and ramification for relative extensions of number fields. It includes a large collection of fundamental classical examples, dealing in particular with relative quadratic extensions as well as relative cyclic extensions of odd prime degree. The unified approach is exclusively algebraic in nature.

• The Exact Hexagon — The Group R⁰(E/F)
• The Group R¹(E/F)
• Some Facts from Class Field Theory
• Determination of R⁰(E/F)
• Determination of R¹(E/F)
• R⁰(E/F), R¹(E/F) for S-Integers
• The Homomorphism C(F) → C(E)
• Unramified Cyclic Extensions
• Ramified Cyclic Extensions
• Relative Quadratic Extensions — Hilbert Symbols
• The Narrow Class Group
• Signs of Units
• CM-Extensions
• The Kernel of C(F) → C(E)
• Units with Almost Independent Signs
• Parity of the Relative Class Number
• Existence of Quadratic Extensions
• Quadratic Extensions of Q — Cyclic 2-Primary Subgroups of C(E)
• Elementary Abelian 2-Primary Subgroups of C(E)
• Imaginary Biquadratic Extensions of Q
• Real Biquadratic Extensions of Q
• Examples
• Non-Abelian Biquadratic Extensions of Q
• The Sets A⁺(2) and A^-(2)
• The 2-Primary Subgroup of K2(0)
• Trivial Galois Action on C(E)