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Darrin Doud

214 TMCB Department of Mathematics Brigham Young University Provo, UT 84602

phone: (801)422-1204 fax: (801)422-0504 e-mail:

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Local corrections of discriminant bounds and small degree extensions of quadratic base fields

with Sharon Brueggeman
International Journal of Number Theory, 4(2008), 349-361.

Abstract: Using analytic techniques of Odlyzko and Poitou, we create tables of lower bounds for discriminants of number fields, including local corrections for ideals of known norm. Comparing the lower bounds found in these tables with upper bounds on discriminants of number fields obtained from calculations involving differents, we prove the nonexistence of a number of small degree extensions of quadratic fields having limited ramification. We note that several of our results require the locally corrected bounds.

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DVI PDF PS

Supplementary Tables and Computer Code

In addition to a preprint of our paper, we also make available further tables of discriminant bounds involving local corrections as well as the Maple code used to construct the tables. These tables consist of unconditional lower bounds for discriminants of number fields of a given degree, with r₁ real places and primes of certain norms. For details about the tables and their construction see the paper.

Unconditional Bounds	Bounds requiring GRH
Degree 5	Degree 5 (GRH)
Degree 6	Degree 6 (GRH)
Degree 7	Degree 7 (GRH)
Degree 8	Degree 8 (GRH)
Degree 9	Degree 9 (GRH)
Degree 10	Degree 10 (GRH)
Degree 11	Degree 11 (GRH)
Degree 12	Degree 12 (GRH)
Degree 13	Degree 13 (GRH)
Degree 14	Degree 14 (GRH)
Degree 15	Degree 15 (GRH)
Degree 16	Degree 16 (GRH)
Degree 17	Degree 17 (GRH)
Degree 18	Degree 18 (GRH)
Degree 19
Degree 20
Degree 22
Degree 24
Degree 25
Degree 26

To download maple code, right click on the links below, and save the file to load into Maple

Maple Code for unconditional case

Maple Code for GRH case (totally complex)

Maple Code for GRH case (two real embeddings)

Cited by

• Patrick Kuhn, Nicolas Robles, and Alexandru Zaharescu, The Larges gap beween zeros of entire L-functions is less than 41.54, J. Math. Anal. Appl., 449 (2017), 1286-1301.
• John W. Jones, Wild ramification bounds and simple group Galois extensions ramified only at 2, Proceedings of the American Mathematical Society, 139 (2011), 807-821.
• Lesseni Sylla, Decic Number Fields with discriminant p^a for p=2,3,5,7, preprint, 2010.
• Benjamin Linowitz, Selectivity in central simple algebras and isospectrality, Ph.D. Thesis, Dartmouth College, 2012
• Mikhail Belolipetsky and Benjamin Linowitz, On Fields of Definition of Arithmetic Kleinian Reflection Groups II, International Mathematics Research Notices, 2014 (2014), 2559-2571.
• Markus Kirschmer, One-class genera of maximal integral quadratic forms, Journal of Number Theory, 136 (2014), 375-393.
• Benjamin Linowitz, Families of mutually isospectral Riemannian orbifolds, Bulletin of the London Mathematical Society, 47 (2015), 47-54.
• Benjamin Linowitz, Matthew Satriano and Roope Vehkalahti, A non-commutative analogue of the Odlyzko bounds and bounds on performance for space-time lattice codes, IEEE Trans. Inform. Theory, 61 (2015), 1971-1984.
• Benjamin Linowitz and John Voight, Small isospectral and nonisometric orbifolds of dimension 2 and 3, Math. Z., 181 (2014), 523-569.
• Markus Kirschmer and David Lorch, Ternary quadratic forms over number fields with small class number, Journal of Number Theory, 161 (2016), 343-361.
• Benjamin Linowitz, D. B. McReynolds, Paul Pollack, and Lola Thompson, Counting and effective rigidity in algebra and geometry, ArXiv preprint, (2014).