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The least inert prime in a real quadratic field


Author: Enrique Treviño
Journal: Math. Comp. 81 (2012), 1777-1797
MSC (2010): Primary 11L40, 11Y40, 11R11
DOI: https://doi.org/10.1090/S0025-5718-2012-02579-8
Published electronically: February 10, 2012
MathSciNet review: 2904602
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Abstract: In this paper, we prove that for any positive fundamental discriminant $ D > 1596$, there is always at least one prime $ p \leq D^{0.45}$ such that the Kronecker symbol $ (D/p) = -1$. This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime $ p$ in a real quadratic field of discriminant $ D > 3705$ is at most $ \sqrt {D}/2$. We use a ``smoothed'' version of the Pólya-Vinogradov inequality, which is very useful for numerically explicit estimates.

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Additional Information

Enrique Treviño
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
Address at time of publication: Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081
Email: etrevin1@swarthmore.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02579-8
Keywords: Character Sums, Pólya–Vinogradov inequality, quadratic fields
Received by editor(s): May 10, 2011
Received by editor(s) in revised form: June 8, 2011
Published electronically: February 10, 2012
Additional Notes: This paper is essentially Chapter 3 of the author’s Ph.D. Dissertation [16].
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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