Dirichlet's unit theorem
The signature of a number field determines the rank of its unit group. Compare real and imaginary quadratic fields, then follow the logarithmic embedding used to prove finite generation.
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Number Field · ring of integers of a number field · abelian group
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Start here: Algebraic Number Theory (v3.08)
James S. Milne · Lecture notes · English · Free full text
Chapter 5 The Unit Theorem, Theorem 5.1 and the logarithmic-embedding proof
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Algebraic Number Theory & Class Field Theory in the library · Preparation and skip guidance · Terminology and translation evidence