The least non-split prime in a number field
Let K be an S n -field ( n ≤ 5 ) with discriminant d K . Let n K be the least prime which does not split completely in K . We prove unconditionally that n K = O ( log | d K | ) , except for O X exp - c log X log log X fields for some constant c > 0 . We also prove that the exceptional set is opti...
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| Published in | Archiv der Mathematik Vol. 117; no. 5; pp. 509 - 513 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.11.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | Let
K
be an
S
n
-field (
n
≤
5
) with discriminant
d
K
. Let
n
K
be the least prime which does not split completely in
K
. We prove unconditionally that
n
K
=
O
(
log
|
d
K
|
)
, except for
O
X
exp
-
c
log
X
log
log
X
fields for some constant
c
>
0
. We also prove that the exceptional set is optimal, and that for any
η
>
0
, the
S
n
-fields such that
n
K
≫
(
log
|
d
K
|
)
1
+
η
are very rare. |
|---|---|
| AbstractList | Let
K
be an
S
n
-field (
n
≤
5
) with discriminant
d
K
. Let
n
K
be the least prime which does not split completely in
K
. We prove unconditionally that
n
K
=
O
(
log
|
d
K
|
)
, except for
O
X
exp
-
c
log
X
log
log
X
fields for some constant
c
>
0
. We also prove that the exceptional set is optimal, and that for any
η
>
0
, the
S
n
-fields such that
n
K
≫
(
log
|
d
K
|
)
1
+
η
are very rare. Let K be an Sn-field (n≤5) with discriminant dK. Let nK be the least prime which does not split completely in K. We prove unconditionally that nK=O(log|dK|), except for OXexp-clogXloglogX fields for some constant c>0. We also prove that the exceptional set is optimal, and that for any η>0, the Sn-fields such that nK≫(log|dK|)1+η are very rare. |
| Author | Kim, Henry H. |
| Author_xml | – sequence: 1 givenname: Henry H. surname: Kim fullname: Kim, Henry H. email: henrykim@math.toronto.edu organization: Department of Mathematics, University of Toronto, Korea Institute for Advanced Study |
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| Cites_doi | 10.1142/S1793042108001432 10.1016/j.jnt.2013.06.009 10.1007/BF02698692 10.1017/S0305004117000019 10.1093/imrn/rny074 |
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| DOI | 10.1007/s00013-021-01650-9 |
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| Keywords | Secondary 11M41 Artin Primary 11R42 Least prime in a conjugacy class Chebotarev density theorem functions |
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| References | CR2 Cho, Kim (CR1) 2013; 133 CR3 Lau, Wu (CR4) 2008; 4 CR5 PJ Cho (1650_CR1) 2013; 133 YK Lau (1650_CR4) 2008; 4 1650_CR5 1650_CR3 1650_CR2 |
| References_xml | – volume: 4 start-page: 423 issue: 3 year: 2008 end-page: 435 ident: CR4 article-title: On the least quadratic non-residue publication-title: Int. J. Number Theory doi: 10.1142/S1793042108001432 contributor: fullname: Wu – ident: CR3 – ident: CR2 – volume: 133 start-page: 4175 year: 2013 end-page: 4187 ident: CR1 article-title: Probabilistic properties of number fields publication-title: J. Number Theory doi: 10.1016/j.jnt.2013.06.009 contributor: fullname: Kim – ident: CR5 – ident: 1650_CR5 doi: 10.1007/BF02698692 – volume: 4 start-page: 423 issue: 3 year: 2008 ident: 1650_CR4 publication-title: Int. J. Number Theory doi: 10.1142/S1793042108001432 contributor: fullname: YK Lau – ident: 1650_CR2 doi: 10.1017/S0305004117000019 – volume: 133 start-page: 4175 year: 2013 ident: 1650_CR1 publication-title: J. Number Theory doi: 10.1016/j.jnt.2013.06.009 contributor: fullname: PJ Cho – ident: 1650_CR3 doi: 10.1093/imrn/rny074 |
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| Snippet | Let
K
be an
S
n
-field (
n
≤
5
) with discriminant
d
K
. Let
n
K
be the least prime which does not split completely in
K
. We prove unconditionally that
n
K
=... Let K be an Sn-field (n≤5) with discriminant dK. Let nK be the least prime which does not split completely in K. We prove unconditionally that nK=O(log|dK|),... |
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| SubjectTerms | Mathematics Mathematics and Statistics Number theory |
| Title | The least non-split prime in a number field |
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