Construction of a certain circular unit and its applications
If \(K\) is number field and \(n\) a natural number, we denote by \(U(K)\) the unit group of the ring of integers of \(K\) and \(\zeta_n\) a primitive \(n\)-th root of unity. Let \(V_n\) be the subgroup of the multiplicative group \(\mathbb{Q}(\zeta_n)\setminus \{0\}\) generated by \(\pm \zeta_n\) and \(1-\zeta_n^{a}\), \(1 \leq a \leq n-1\). The group of cyclotomic units \(C(\mathbb{Q}(\zeta_n))\) of \(\mathbb{Q}(\zeta_n)\) is defined by: \(C(\mathbb{Q}(\zeta_n))=U(\mathbb{Q}(\zeta_n))\cap V_n\). Let \(K/\mathbb{Q}\) be a finite abelian extension having conductor \(n\) (by Kronecker-Weber theorem, the smallest natural number such that \(k\subset\mathbb{Q}(\zeta_n)\)). Let \(W_n\) be the subgroup of the multiplicative group \(\mathbb{Q}(\zeta_n)\setminus \{0\}\) generated by \(-1\) and the elements \(N_{\mathbb{Q}(\zeta_m)/K\cap \mathbb{Q}(\zeta_m)}(1-\zeta_m^{a})\), where \(m\not=1\) is a divisor of \(n\) relatively prime to \(a\). In [Invent. Math. 62, 181--234 (1980; Zbl 0465.12001)], \textit{W. Sinnott} defined the group \(C_S(K)\) of circular units of \(K\) by: \(C_S(K)=U(K)\cap W_n\). In [Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, NY: Springer (1997; Zbl 0966.11047)], \textit{L. Washington} defined \(C_W(K)=U(K)\cap C(\mathbb{Q}(\zeta_n))\). It is immediate that \(C_S(K)\) is a subgroup of \(C_W(K)\). Let \(n=p_1^{e_1}p_2^{e_2}\), where the \(p_i\), \(1 \leq i \leq 2\), are distinct prime odd numbers, and the \(e_i\), \(1 \leq i \leq 2\), are natural numbers. Let \(k/\mathbb{Q}\) be a finite real abelian extension with conductor \(n\). Assume that \(k\) is contained in the compositum of the maximal real subfields of \(\mathbb{Q}(\zeta_{p_1^{e_1} })\) and \(\mathbb{Q}(\zeta_{p_2^{e_2} })\) (\(\mathbb{Q}(\zeta_{p_1^{e_1}} +\zeta_{p_1^{e_1}}^{-1})\) and \(\mathbb{Q}(\zeta_{p_2^{e_2}} +\zeta_{p_2^{e_2}}^{-1})\) ). The main result of the paper is the following. The authors construct an element \(\theta\) in \(C_W(k)\) with non trivial class in the quotient group \(C_W(k)/C_S(k)\). As an application they obtain divisors of the class number of \(k\) for some \(k\). An example is the following. Put \(e_i=1\) for \(1 \leq i \leq 2\), \([k:\mathbb{Q}]=s\), \(l_1=(p_1-1)/\text{ord}_{p_1}(p_2)\) and \(l_2=(p_2-1)/\text{ord}_{p_2}(p_1)\), where \(\text{ord}_{p_i}(p_j)\) for \(i\not=j\) is the order of the class of \(p_j\) in the multiplicative group \((\mathbb{Z}/p_i\mathbb{Z})\setminus \{0\}\). Let \(d=\gcd(l_1, l_2, s)\). Suppose that \(k/\mathbb{Q}\) and \(\mathbb{Q}(\zeta_{p_1})/\mathbb{Q}\) (resp. \(\mathbb{Q}(\zeta_{p_2})/\mathbb{Q}\)) are linearly disjoint. Then the order of \(\theta\) is \(d\) and the class number of \(k\) is divisible by \(ds\). Also, as another application to their main result, the authors prove a known result concerning the parity class number for certain quadratic number fields.
scientific article; zbMATH DE number 5859977