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Let $\sigma_1,\ldots,\sigma_{r_1}$ be the real embeddings of a number field $K$ into the complex numbers $\mathbb{C}$, and $\sigma_{r_1+ 1},\ldots,\sigma_{r_1+r_2}$ be complex embeddings of $K$ into $\C$ such that no two are complex conjugate. Let $u_1,\ldots,u_r$ be a set of fundamental units of $K$. Then $r = r_1 + r_2 -1$.

The regulator of $K$ is a positive real number defined as the absolute value of the determinant of the matrix $(d_i\log{ \sigma_j(u_i)})$, where any one column is removed and $d_i$ is the degree of the embedding $\sigma_i$.

Knowl status:
  • Review status: reviewed
  • Last edited by Alina Bucur on 2018-07-07 23:18:32
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