Monogenity of totally real algebraic extension fields over a cyclotomic field

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• 作者：Nadia Khan^a ; ¹ ; ^{p109958@nu.edu.pk" class="auth_mail" title="E-mail the corresponding author} ; Shin-ichi Katayama^b ; ^{katayama@ias.tokushima-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; Toru Nakahara^a ; ¹ ; ^{toru.nakahara@nu.edu.pk" class="auth_mail" title="E-mail the corresponding author} ; ^{nakahara@ms.saga-u.ac.jp" class="auth_mail" title="E-mail the corresponding author} ; ^{toru.nakahara@qu.edu.pk" class="auth_mail" title="E-mail the corresponding author} ; Tsuyoshi Uehara^c ; ^{uehara@ma.is.saga-u.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11R04 ; 11R21 ; 11R29
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：158
• 期：Complete
• 页码：348-355
• 全文大小：299 K

文摘

Let K   be a composite field of a cyclotomic field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=df7c9c31bfe5f0088572c3d75c515f97" title="Click to view the MathML source">k_nclass="mathContainer hidden">class="mathCode">$k_n$ of odd conductor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si2.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=d4ccf034b5a6f9be253f0d47bafd9070" title="Click to view the MathML source">n鈮?class="mathContainer hidden">class="mathCode">$n鈮?/mo><mn>3</mn>$ or even one 鈮? with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si3.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=eb8302f0669fdec9d9457154313e3334" title="Click to view the MathML source">4|nclass="mathContainer hidden">class="mathCode">$4|n$ and a totally real algebraic extension field F over the rationals class="boldFont">Q   and both fields class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=df7c9c31bfe5f0088572c3d75c515f97" title="Click to view the MathML source">k_nclass="mathContainer hidden">class="mathCode">$k_n$ and F are linearly disjoint over class="boldFont">Q to each other. Then the purpose of this paper is to prove that such a relatively totally real extension field K   over a cyclotomic field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si1.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=df7c9c31bfe5f0088572c3d75c515f97" title="Click to view the MathML source">k_nclass="mathContainer hidden">class="mathCode">$k_n$ has no power integral basis. Each of the composite fields K   is also a CM field over the maximal real subfield class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si4.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=2aba2f4dc569ba494645ab237addb4c3">class="imgLazyJSB inlineImage" height="19" width="42" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002310-si4.gif">class="mathContainer hidden">class="mathCode">$k_n^{+}\cdot F$ of K  . This result involves the previous work for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si5.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=d3b3e680f015669e19fa0177dfadb2fb" title="Click to view the MathML source">K=k_n⋅Fclass="mathContainer hidden">class="mathCode">$K=k_n\cdot F$ of the Eisenstein field class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si6.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=bedfa1c2330d545c395a9d69996e6995" title="Click to view the MathML source">k_n=k₃class="mathContainer hidden">class="mathCode">$k_n=k_3$ and the maximal real subfields class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si7.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=4fc488b51e6dd836cf3188cb65db4c1d">class="imgLazyJSB inlineImage" height="22" width="56" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002310-si7.gif">class="mathContainer hidden">class="mathCode">$F=k_{p^n}^{+}$ of prime power conductor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si37.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=a646beab36529bc1ec37184456900993" title="Click to view the MathML source">pⁿclass="mathContainer hidden">class="mathCode">$p^n$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si38.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=3317cd00d4f5b2d196316069e6bb1860" title="Click to view the MathML source">p鈮?class="mathContainer hidden">class="mathCode">$p鈮?/mo><mn>5</mn>$, and an analogue class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si5.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=d3b3e680f015669e19fa0177dfadb2fb" title="Click to view the MathML source">K=k_n⋅Fclass="mathContainer hidden">class="mathCode">$K=k_n\cdot F$ of cyclotomic fields class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si10.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=722595acd67032a59c285a0695ef1033">class="imgLazyJSB inlineImage" height="18" width="124" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002310-si10.gif">class="mathContainer hidden">class="mathCode"> with a totally real algebraic fields F   of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si11.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=29d48eead58dc2afcd6c8b1b45611c54" title="Click to view the MathML source">K=k₄⋅Fclass="mathContainer hidden">class="mathCode">$K=k_4\cdot F$ with a cyclic cubic field F   except for class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si12.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=6a1eabd45ecc1f8e00598740e15de7b2">class="imgLazyJSB inlineImage" height="21" width="44" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002310-si12.gif">class="mathContainer hidden">class="mathCode">$k_4\cdot k_7^{+}$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X15002310&_mathId=si13.gif&_user=111111111&_pii=S0022314X15002310&_rdoc=1&_issn=0022314X&md5=d2283f2d67b497fbb21f8eb6485c5ba9">class="imgLazyJSB inlineImage" height="22" width="46" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X15002310-si13.gif">class="mathContainer hidden">class="mathCode">$k_4\cdot k_{3^2}^{+}$ of conductors 28 and 36.