4 The prime-conductor index theorem
The index-theorem subgroup is the normalized cyclotomic-unit subgroup.
The subgroup used in the index theorem agrees with the subgroup \(C^+\) used in the saturation argument.
The proof gives both inclusions. One direction shows that the normalized generators lie in the subgroup generated by the standard real cyclotomic units; the other direction shows that the standard generators are obtained from the normalized family.
The deleted Fourier determinant calculation supplies the Kummer–Dirichlet determinant required by the Sinnott index pipeline.
The proof composes the deleted Fourier determinant identity with the regulator identity for the cyclotomic-unit family, producing the exact ‘KummerDirichletDeterminant‘ hypothesis needed downstream.
For \(p\ge 5\), the normalized cyclotomic-unit index has the same \(p\)-primary divisibility as \(h^+\):
The deleted Fourier determinant gives the Kummer–Dirichlet determinant. Sinnott’s prime-conductor index theorem applies to the corresponding index subgroup. Theorem 4.2 identifies that subgroup with \(C^+\), and theorem 1.6 passes between the standard and normalized indices at the odd prime \(p\).
For every odd prime conductor,
If \(p=3\), the normalized subgroup is the whole unit group and \(h^+=1\), so both divisibility statements are false. If \(p\ge 5\), the theorem is theorem 4.4.