Dirichlet’s Unit Theorem
نویسنده
چکیده
Theorem 1.1 (Dirichlet, 1846). Let K be a number field with r1 real embeddings and 2r2 pairs of complex conjugate embeddings. The unit group of any order in K is finitely generated with r1 + r2 − 1 independent generators of infinite order. More precisely, letting r = r1 + r2 − 1, any order O in K contains multiplicatively independent units ε1, . . . , εr of infinite order such that every unit in O can be written uniquely in the form ζε1 1 · · · εr r , where ζ is a root of unity in O and the mi’s are in Z. Abstractly, O× ∼= μ(O)× Zr1+r2−1, where μ(O) is the finite cyclic group of roots of unity in O.
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