Formulas for Chebotarev densities of Galois extensions of number fields View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2019-03

AUTHORS

Naomi Sweeting, Katharine Woo

ABSTRACT

We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436–451, 1977) to the setting of arbitrary finite Galois extensions of number fields L / K. In particular, if C⊂G=Gal(L/K) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K: -limX→∞∑2≤N(I)≤XI∈S(L/K;C)μK(I)N(I)=|C||G|,where μK(I) denotes the generalized Möbius function and S(L / K; C) is the set of ideals I⊂OK such that I has a unique prime divisor p of minimal norm and the Artin symbol L/Kp is C. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi’s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields. More... »

PAGES

4

References to SciGraph publications

  • 1992-12. An interval result for the number field ψ(x,y) function in MANUSCRIPTA MATHEMATICA
  • 1903-12. Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes in MATHEMATISCHE ANNALEN
  • 2017-12. A new formula for Chebotarev Densities in RESEARCH IN NUMBER THEORY
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s40993-018-0142-x

    DOI

    http://dx.doi.org/10.1007/s40993-018-0142-x

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1109934019


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "University of Chicago", 
              "id": "https://www.grid.ac/institutes/grid.170205.1", 
              "name": [
                "Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Sweeting", 
            "givenName": "Naomi", 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Stanford University", 
              "id": "https://www.grid.ac/institutes/grid.168010.e", 
              "name": [
                "Department of Mathematics, Stanford University, 94305, Stanford, CA, USA"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Woo", 
            "givenName": "Katharine", 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "https://doi.org/10.1016/j.exmath.2006.08.001", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003455399"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02567771", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021994029", 
              "https://doi.org/10.1007/bf02567771"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02567771", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021994029", 
              "https://doi.org/10.1007/bf02567771"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01444310", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1030940432", 
              "https://doi.org/10.1007/bf01444310"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1002/cpa.3160020401", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039879014"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/0022-314x(77)90005-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1047167107"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s40993-017-0093-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1091014854", 
              "https://doi.org/10.1007/s40993-017-0093-7"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2019-03", 
        "datePublishedReg": "2019-03-01", 
        "description": "We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436\u2013451, 1977) to the setting of arbitrary finite Galois extensions of number fields L / K. In particular, if C\u2282G=Gal(L/K) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K: -limX\u2192\u221e\u22112\u2264N(I)\u2264XI\u2208S(L/K;C)\u03bcK(I)N(I)=|C||G|,where \u03bcK(I) denotes the generalized M\u00f6bius function and S(L / K; C) is the set of ideals I\u2282OK such that I has a unique prime divisor p of minimal norm and the Artin symbol L/Kp is C. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi\u2019s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.", 
        "genre": "non_research_article", 
        "id": "sg:pub.10.1007/s40993-018-0142-x", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": true, 
        "isFundedItemOf": [
          {
            "id": "sg:grant.4893317", 
            "type": "MonetaryGrant"
          }
        ], 
        "isPartOf": [
          {
            "id": "sg:journal.1053185", 
            "issn": [
              "2522-0160", 
              "2363-9555"
            ], 
            "name": "Research in Number Theory", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "5"
          }
        ], 
        "name": "Formulas for Chebotarev densities of Galois extensions of number fields", 
        "pagination": "4", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "371b7631bd9ed6232fe51d2e4ab3ae10951f439602fda2c6205e1a824385d2bc"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s40993-018-0142-x"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1109934019"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s40993-018-0142-x", 
          "https://app.dimensions.ai/details/publication/pub.1109934019"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-11T08:23", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000293_0000000293/records_12008_00000000.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://link.springer.com/10.1007%2Fs40993-018-0142-x"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s40993-018-0142-x'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s40993-018-0142-x'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s40993-018-0142-x'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s40993-018-0142-x'


     

    This table displays all metadata directly associated to this object as RDF triples.

    92 TRIPLES      21 PREDICATES      33 URIs      19 LITERALS      7 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s40993-018-0142-x schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N021b5f683cd243d4a8fafad2a4209c1a
    4 schema:citation sg:pub.10.1007/bf01444310
    5 sg:pub.10.1007/bf02567771
    6 sg:pub.10.1007/s40993-017-0093-7
    7 https://doi.org/10.1002/cpa.3160020401
    8 https://doi.org/10.1016/0022-314x(77)90005-1
    9 https://doi.org/10.1016/j.exmath.2006.08.001
    10 schema:datePublished 2019-03
    11 schema:datePublishedReg 2019-03-01
    12 schema:description We generalize the Chebotarev density formulas of Dawsey (Res Number Theory 3:27, 2017) and Alladi (J Number Theory 9:436–451, 1977) to the setting of arbitrary finite Galois extensions of number fields L / K. In particular, if C⊂G=Gal(L/K) is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of K: -limX→∞∑2≤N(I)≤XI∈S(L/K;C)μK(I)N(I)=|C||G|,where μK(I) denotes the generalized Möbius function and S(L / K; C) is the set of ideals I⊂OK such that I has a unique prime divisor p of minimal norm and the Artin symbol L/Kp is C. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi’s concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.
    13 schema:genre non_research_article
    14 schema:inLanguage en
    15 schema:isAccessibleForFree true
    16 schema:isPartOf N743c92683bec4c2bb8d232371d19b30a
    17 N885770b3c9334f8ba6236df43cf51138
    18 sg:journal.1053185
    19 schema:name Formulas for Chebotarev densities of Galois extensions of number fields
    20 schema:pagination 4
    21 schema:productId N4418a5b877d14c28a13c16460cd8e771
    22 N561d41ba4ad744dbb660e5a05fb0b79e
    23 N677d9556aa2644e8a79e12bf43161ba2
    24 schema:sameAs https://app.dimensions.ai/details/publication/pub.1109934019
    25 https://doi.org/10.1007/s40993-018-0142-x
    26 schema:sdDatePublished 2019-04-11T08:23
    27 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    28 schema:sdPublisher Nced523fa996f4ed68cb688cf5a7dbb1a
    29 schema:url https://link.springer.com/10.1007%2Fs40993-018-0142-x
    30 sgo:license sg:explorer/license/
    31 sgo:sdDataset articles
    32 rdf:type schema:ScholarlyArticle
    33 N021b5f683cd243d4a8fafad2a4209c1a rdf:first N32bc6b498e3d4570ab647f50f6b4185c
    34 rdf:rest N8fc96686a6104681bda81f81332437f3
    35 N32bc6b498e3d4570ab647f50f6b4185c schema:affiliation https://www.grid.ac/institutes/grid.170205.1
    36 schema:familyName Sweeting
    37 schema:givenName Naomi
    38 rdf:type schema:Person
    39 N4418a5b877d14c28a13c16460cd8e771 schema:name doi
    40 schema:value 10.1007/s40993-018-0142-x
    41 rdf:type schema:PropertyValue
    42 N48a46aa616724188bf0f16fb5264f56b schema:affiliation https://www.grid.ac/institutes/grid.168010.e
    43 schema:familyName Woo
    44 schema:givenName Katharine
    45 rdf:type schema:Person
    46 N561d41ba4ad744dbb660e5a05fb0b79e schema:name dimensions_id
    47 schema:value pub.1109934019
    48 rdf:type schema:PropertyValue
    49 N677d9556aa2644e8a79e12bf43161ba2 schema:name readcube_id
    50 schema:value 371b7631bd9ed6232fe51d2e4ab3ae10951f439602fda2c6205e1a824385d2bc
    51 rdf:type schema:PropertyValue
    52 N743c92683bec4c2bb8d232371d19b30a schema:issueNumber 1
    53 rdf:type schema:PublicationIssue
    54 N885770b3c9334f8ba6236df43cf51138 schema:volumeNumber 5
    55 rdf:type schema:PublicationVolume
    56 N8fc96686a6104681bda81f81332437f3 rdf:first N48a46aa616724188bf0f16fb5264f56b
    57 rdf:rest rdf:nil
    58 Nced523fa996f4ed68cb688cf5a7dbb1a schema:name Springer Nature - SN SciGraph project
    59 rdf:type schema:Organization
    60 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    61 schema:name Mathematical Sciences
    62 rdf:type schema:DefinedTerm
    63 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    64 schema:name Pure Mathematics
    65 rdf:type schema:DefinedTerm
    66 sg:grant.4893317 http://pending.schema.org/fundedItem sg:pub.10.1007/s40993-018-0142-x
    67 rdf:type schema:MonetaryGrant
    68 sg:journal.1053185 schema:issn 2363-9555
    69 2522-0160
    70 schema:name Research in Number Theory
    71 rdf:type schema:Periodical
    72 sg:pub.10.1007/bf01444310 schema:sameAs https://app.dimensions.ai/details/publication/pub.1030940432
    73 https://doi.org/10.1007/bf01444310
    74 rdf:type schema:CreativeWork
    75 sg:pub.10.1007/bf02567771 schema:sameAs https://app.dimensions.ai/details/publication/pub.1021994029
    76 https://doi.org/10.1007/bf02567771
    77 rdf:type schema:CreativeWork
    78 sg:pub.10.1007/s40993-017-0093-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1091014854
    79 https://doi.org/10.1007/s40993-017-0093-7
    80 rdf:type schema:CreativeWork
    81 https://doi.org/10.1002/cpa.3160020401 schema:sameAs https://app.dimensions.ai/details/publication/pub.1039879014
    82 rdf:type schema:CreativeWork
    83 https://doi.org/10.1016/0022-314x(77)90005-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1047167107
    84 rdf:type schema:CreativeWork
    85 https://doi.org/10.1016/j.exmath.2006.08.001 schema:sameAs https://app.dimensions.ai/details/publication/pub.1003455399
    86 rdf:type schema:CreativeWork
    87 https://www.grid.ac/institutes/grid.168010.e schema:alternateName Stanford University
    88 schema:name Department of Mathematics, Stanford University, 94305, Stanford, CA, USA
    89 rdf:type schema:Organization
    90 https://www.grid.ac/institutes/grid.170205.1 schema:alternateName University of Chicago
    91 schema:name Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA
    92 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...