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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 30, Issue 3

# The least unramified prime which does not split completely

Asif Zaman
Published Online: 2017-08-30 | DOI: https://doi.org/10.1515/forum-2017-0081

## Abstract

Let $K/F$ be a finite extension of number fields of degree $n\ge 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal $𝔭$ of F which is degree 1 over $ℚ$ and does not ramify or split completely in K. We improve upon the previous best known general estimates due to Li [7] when $F=ℚ$, and Murty and Patankar [9] when $K/F$ is Galois. Our bounds are the first when $K/F$ is not assumed to be Galois and $F\ne ℚ$.

Keywords: Non-split prime; prime ideals

MSC 2010: 11R44; 11R42

## References

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J. Thorner and A. Zaman, An explicit bound for the least prime ideal in the Chebotarev density theorem, Algebra Number Theory 11 (2017), no. 5, 1135–1197.

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Revised: 2017-08-09

Published Online: 2017-08-30

Published in Print: 2018-05-01

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 651–661, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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