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Biquadratic fields having a non-principal Euclidean ideal class. (English) Zbl 1482.11147

Extending work of C. Hsu [Int. J. Number Theory 12, No. 4, 1123–1136 (2016; Zbl 1415.11159)], the authors prove the biquadratic number fields \(K = \mathbb{Q}(\sqrt{q}, \sqrt{kr})\), where \(k \equiv r \equiv 1 \bmod 4\) and \(q = 2\) or \(q \equiv 3 \bmod 4\) are primes, have a Euclidean ideal class if \(K\) has class number \(2\). We remark that the construction of the Hilbert class field of \(K\) would have followed painlessly from genus theory applied to the quadratic number field \(\mathbb{Q}(\sqrt{qkr})\).

MSC:

11R29 Class numbers, class groups, discriminants
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11R16 Cubic and quartic extensions

Citations:

Zbl 1415.11159
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References:

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