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Imaginary cyclic fields of degree \(p-1\) whose ideal class groups have \(p\)-rank at least two. (English) Zbl 1294.11199

Summary: Let \(p\) be a prime number which is congruent to 3 modulo 4. For an odd positive integer \(n\), we define a quadratic field \(k_{n,p}\) by \(k_{n,p} := \mathbb Q(\sqrt{4 - p^{pn}}\,)\). Moreover let \(M_{p,n}\) be the composite field of \(k_{n,p}\) and the maximal real subfield of the \(p\)th cyclotomic field. Then \(M_{p,n}\) is an imaginary cyclic field of degree \(p-1\). In this paper, we prove that the \(p\)-rank of ideal class groups of \(M_{p,n}\) is at least 2 for any odd integer \( n \geq 1 \) except for \((p,n) = (3, 1)\). Furthermore, we can show \(M_{p,n} \neq M_{p,m}\) for any distinct two integers \(n\) and \(m\). As a consequence, we see that there exist infinitely many imaginary cyclic fields of degree \(p- 1\) whose ideal class group have \(p\)-rank at least 2.

MSC:

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
11D61 Exponential Diophantine equations
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