Abstract
Let q be a power of an odd prime number \({p, k=\mathbb{F}_{q}(t)}\) be the rational function field over the finite field \({\mathbb{F}_{q}.}\) In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields.
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Li, Y., Hu, S. Capitulation problem for global function fields. Arch. Math. 97, 413–421 (2011). https://doi.org/10.1007/s00013-011-0326-2
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DOI: https://doi.org/10.1007/s00013-011-0326-2