4 The prime-conductor index theorem
The subgroup featuring in the prime-conductor index theorem is the normalised cyclotomic-unit subgroup \(C^+_{\mathrm{norm}}\) of definition 1.4.
Up to the normalisation of theorem 1.6, the subgroup used in the index theorem agrees with the subgroup \(C^+\) used in the saturation argument: each contains representatives generating the other modulo \(\{ \pm 1\} \) and squares.
Both inclusions are checked on generators. A normalised generator \(\xi _a\) times a root of unity recovers the cyclotomic unit \((1-\zeta _p^a)/(1-\zeta _p)\), whose product with its conjugate is \(\varepsilon _a = \xi _a^2\); conversely each \(\varepsilon _a\) is the square of \(\xi _a\), so the two subgroups generate each other up to squares and signs.
The deleted-Fourier determinant identity supplies the Kummer–Dirichlet determinant hypothesis required by the Sinnott index pipeline: the regulator-type determinant of the cyclotomic-unit family is, up to an explicit nonzero factor, a product of the character sums \(\sum _a \chi (a) \log |1-e^{2\pi i a/p}|\) over the even nontrivial characters.
The logarithmic embeddings of the cyclotomic units form a matrix whose group-theoretic structure (rows and columns indexed by a cyclic group) is diagonalised by the discrete Fourier transform with one character deleted. Composing the resulting determinant identity with the regulator identity for the cyclotomic-unit family produces exactly the Kummer–Dirichlet determinant needed downstream.
For \(p\ge 5\), the normalised cyclotomic-unit index has the same \(p\)-divisibility as \(h^+\):
This is Sinnott’s prime-conductor index theorem: for prime conductor the index of the circular units in the full unit group equals \(h^+\). The deleted-Fourier determinant of theorem 4.3 provides the determinant computation at the heart of the proof, theorem 4.2 identifies the index subgroup with the one used here, and theorem 1.6 transfers \(p\)-divisibility between the standard and normalised indices.
For every odd prime conductor,
For \(p=3\) the field \(K^+\) is \(\mathbb {Q}\): the normalised subgroup is the whole unit group \(\{ \pm 1\} \) and \(h^+=1\), so both sides are false. For \(p\ge 5\) this is theorem 4.4.