1 The real cyclotomic-unit subgroup

Definition 1.1

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For \(2\le a\le (p-1)/2\), the real cyclotomic unit \(\varepsilon _a\) is the unit of \(\mathcal O_{K^+}\) obtained by descending the product of the cyclotomic unit \((1-\zeta _p^a)/(1-\zeta _p)\) of \(\mathcal O_K\) with its complex conjugate:

\[ \varepsilon _a \; =\; \frac{(1-\zeta _p^a)\, (1-\zeta _p^{-a})}{(1-\zeta _p)\, (1-\zeta _p^{-1})} . \]

This element is fixed by complex conjugation, hence lies in \((\mathcal O_{K^+})^\times \).

Definition 1.2

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The standard generators are the \((p-3)/2\) real cyclotomic units

\[ \varepsilon _2,\ \varepsilon _3,\ \ldots ,\ \varepsilon _{(p-1)/2} \]

of definition 1.1, indexed in the formalisation by \(i \in \{ 0, \ldots , (p-5)/2\} \) via \(a = i + 2\).

Definition 1.3

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The subgroup \(C^+\subseteq (\mathcal O_{K^+})^\times \) is generated by \(-1\) and the standard real cyclotomic units:

\[ C^+=\bigl\langle -1,\ \varepsilon _2,\ldots ,\varepsilon _{(p-1)/2} \bigr\rangle . \]

Definition 1.4

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For \(2 \le a \le (p-1)/2\), the normalised cyclotomic unit is

\[ \xi _a \; =\; \zeta _p^{\, (1-a)/2}\, \frac{1-\zeta _p^a}{1-\zeta _p}, \]

where the exponent \((1-a)/2\) is read modulo \(p\). The twist by the root of unity makes \(\xi _a\) fixed by complex conjugation, so it defines a unit of \(\mathcal O_{K^+}\). The normalised subgroup \(C^+_{\mathrm{norm}}\) is generated by \(-1\) and the units \(\xi _a\).

Theorem 1.5

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Each normalised generator squares to the corresponding standard generator:

\[ \xi _a^{\, 2} = \varepsilon _a \qquad (2 \le a \le (p-1)/2). \]

Proof ▶

Using the identity \(1-\zeta _p^{-a} = -\zeta _p^{-a}(1-\zeta _p^a)\) in the numerator and denominator of \(\varepsilon _a\),

\[ \varepsilon _a = \frac{(1-\zeta _p^a)\cdot \bigl(-\zeta _p^{-a}\bigr)(1-\zeta _p^a)}{(1-\zeta _p)\cdot \bigl(-\zeta _p^{-1}\bigr)(1-\zeta _p)} = \zeta _p^{\, 1-a}\, \frac{(1-\zeta _p^a)^2}{(1-\zeta _p)^2} = \xi _a^{\, 2}. \]

Theorem 1.6

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For an odd prime \(p\),

\[ p\mid [(\mathcal O_{K^+})^\times :C^+] \Longleftrightarrow p\mid [(\mathcal O_{K^+})^\times :C^+_{\mathrm{norm}}]. \]

Proof ▶

By theorem 1.5, \(C^+ \subseteq C^+_{\mathrm{norm}}\) and every square of a normalised generator lies in \(C^+\), so the quotient \(C^+_{\mathrm{norm}}/C^+\) is an elementary abelian \(2\)-group of bounded rank; in particular the two indices differ by a power of \(2\). Since \(p\) is odd, multiplying or dividing by a power of \(2\) does not affect \(p\)-divisibility.