Normal integral basis of an unramified quadratic extension over a cyclotomic ℤ₂-extension
Let ℓ be an odd prime number. LetK/ℚ be a real cyclic extension of degree ℓ,AK the 2-part of the ideal class group ofK, andH/Kthe class field corresponding to A K / A K 2 . LetKn be thenth layer of the cyclotomic ℤ₂-extension overK. We consider the questions (Q1) “doesH/Khas a normal integral basis?...
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| Veröffentlicht in: | Journal de theorie des nombres de bordeaux 2016-01, Vol.28 (2), p.325-345 |
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| container_issue | 2 |
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| container_title | Journal de theorie des nombres de bordeaux |
| container_volume | 28 |
| creator | ICHIMURA, Humio SUMIDA-TAKAHASHI, Hiroki |
| description | Let ℓ be an odd prime number. LetK/ℚ be a real cyclic extension of degree ℓ,AK
the 2-part of the ideal class group ofK, andH/Kthe class field corresponding to
A
K
/
A
K
2
. LetKn
be thenth layer of the cyclotomic ℤ₂-extension overK. We consider the questions (Q1) “doesH/Khas a normal integral basis?”, and (Q2) “if not, does the pushed-up extensionHKn/Kn
has a normal integral basis for somen≥ 1?” Under some assumptions on ℓ andK, we answer these questions in terms of the 2-adicL-function associated to the base fieldK. We also give some numerical examples. |
| doi_str_mv | 10.5802/jtnb.942 |
| format | Article |
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the 2-part of the ideal class group ofK, andH/Kthe class field corresponding to
A
K
/
A
K
2
. LetKn
be thenth layer of the cyclotomic ℤ₂-extension overK. We consider the questions (Q1) “doesH/Khas a normal integral basis?”, and (Q2) “if not, does the pushed-up extensionHKn/Kn
has a normal integral basis for somen≥ 1?” Under some assumptions on ℓ andK, we answer these questions in terms of the 2-adicL-function associated to the base fieldK. We also give some numerical examples.</description><identifier>ISSN: 1246-7405</identifier><identifier>EISSN: 2118-8572</identifier><identifier>DOI: 10.5802/jtnb.942</identifier><language>eng</language><publisher>cedram</publisher><subject>Algebra ; Integers ; Mathematical congruence ; Mathematical integrals ; Mathematical rings ; Numbers ; Polynomials ; Power series ; Prime numbers</subject><ispartof>Journal de theorie des nombres de bordeaux, 2016-01, Vol.28 (2), p.325-345</ispartof><rights>Société Arithmétique de Bordeaux, 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/26274035$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/26274035$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,803,832,27924,27925,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>ICHIMURA, Humio</creatorcontrib><creatorcontrib>SUMIDA-TAKAHASHI, Hiroki</creatorcontrib><title>Normal integral basis of an unramified quadratic extension over a cyclotomic ℤ₂-extension</title><title>Journal de theorie des nombres de bordeaux</title><description>Let ℓ be an odd prime number. LetK/ℚ be a real cyclic extension of degree ℓ,AK
the 2-part of the ideal class group ofK, andH/Kthe class field corresponding to
A
K
/
A
K
2
. LetKn
be thenth layer of the cyclotomic ℤ₂-extension overK. We consider the questions (Q1) “doesH/Khas a normal integral basis?”, and (Q2) “if not, does the pushed-up extensionHKn/Kn
has a normal integral basis for somen≥ 1?” Under some assumptions on ℓ andK, we answer these questions in terms of the 2-adicL-function associated to the base fieldK. We also give some numerical examples.</description><subject>Algebra</subject><subject>Integers</subject><subject>Mathematical congruence</subject><subject>Mathematical integrals</subject><subject>Mathematical rings</subject><subject>Numbers</subject><subject>Polynomials</subject><subject>Power series</subject><subject>Prime numbers</subject><issn>1246-7405</issn><issn>2118-8572</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNo9j01KAzEAhYMoWKvgBYRcIDW_k8xSin9QdKNLKUkmkQwziSap2G3xKJ6sJ3FAcfUefI8PHgDnBC-EwvSyr9EsWk4PwIwSopASkh6CGaG8QZJjcQxOSukxpqxp1Qy8PKQ86gGGWN1rnorRJRSYPNQRbmLWY_DBdfB9o7usa7DQfVYXS0gRpg-XoYZ2a4dU0zix_df3frdD_5NTcOT1UNzZX87B88310_IOrR5v75dXK9QTKSvyHZGUOG2MwKptlFeYEIatd84Jw1nnWmJb3BjOjZZSME8tlpx53VFlbMPm4OLX25ea8voth1Hn7Zo2dLrMBPsB8iJVdA</recordid><startdate>20160101</startdate><enddate>20160101</enddate><creator>ICHIMURA, Humio</creator><creator>SUMIDA-TAKAHASHI, Hiroki</creator><general>cedram</general><scope/></search><sort><creationdate>20160101</creationdate><title>Normal integral basis of an unramified quadratic extension over a cyclotomic ℤ₂-extension</title><author>ICHIMURA, Humio ; SUMIDA-TAKAHASHI, Hiroki</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-j177t-fd1721eabb508968f801130cfeee5b43de91c906b44ba7753f2c0743fad28bc63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Algebra</topic><topic>Integers</topic><topic>Mathematical congruence</topic><topic>Mathematical integrals</topic><topic>Mathematical rings</topic><topic>Numbers</topic><topic>Polynomials</topic><topic>Power series</topic><topic>Prime numbers</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>ICHIMURA, Humio</creatorcontrib><creatorcontrib>SUMIDA-TAKAHASHI, Hiroki</creatorcontrib><jtitle>Journal de theorie des nombres de bordeaux</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>ICHIMURA, Humio</au><au>SUMIDA-TAKAHASHI, Hiroki</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Normal integral basis of an unramified quadratic extension over a cyclotomic ℤ₂-extension</atitle><jtitle>Journal de theorie des nombres de bordeaux</jtitle><date>2016-01-01</date><risdate>2016</risdate><volume>28</volume><issue>2</issue><spage>325</spage><epage>345</epage><pages>325-345</pages><issn>1246-7405</issn><eissn>2118-8572</eissn><abstract>Let ℓ be an odd prime number. LetK/ℚ be a real cyclic extension of degree ℓ,AK
the 2-part of the ideal class group ofK, andH/Kthe class field corresponding to
A
K
/
A
K
2
. LetKn
be thenth layer of the cyclotomic ℤ₂-extension overK. We consider the questions (Q1) “doesH/Khas a normal integral basis?”, and (Q2) “if not, does the pushed-up extensionHKn/Kn
has a normal integral basis for somen≥ 1?” Under some assumptions on ℓ andK, we answer these questions in terms of the 2-adicL-function associated to the base fieldK. We also give some numerical examples.</abstract><pub>cedram</pub><doi>10.5802/jtnb.942</doi><tpages>21</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1246-7405 |
| ispartof | Journal de theorie des nombres de bordeaux, 2016-01, Vol.28 (2), p.325-345 |
| issn | 1246-7405 2118-8572 |
| language | eng |
| recordid | cdi_jstor_primary_26274035 |
| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing; EZB-FREE-00999 freely available EZB journals |
| subjects | Algebra Integers Mathematical congruence Mathematical integrals Mathematical rings Numbers Polynomials Power series Prime numbers |
| title | Normal integral basis of an unramified quadratic extension over a cyclotomic ℤ₂-extension |
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