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 k_n$k_n$ of odd conductor n鈮?$n鈮?/mo><mn>3</mn>$ or even one 鈮? with 4|n$4|n$ and a totally real algebraic extension field F over the rationals Q   and both fields k_n$k_n$ and F are linearly disjoint over 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 k_n$k_n$ has no power integral basis. Each of the composite fields K   is also a CM field over the maximal real subfield $k_n^{+}\cdot F$ of K  . This result involves the previous work for K=k_n⋅F$K=k_n\cdot F$ of the Eisenstein field k_n=k₃$k_n=k_3$ and the maximal real subfields $F=k_{p^n}^{+}$ of prime power conductor pⁿ$p^n$ with p鈮?$p鈮?/mo><mn>5</mn>$, and an analogue K=k_n⋅F$K=k_n\cdot F$ of cyclotomic fields with a totally real algebraic fields F   of K=k₄⋅F$K=k_4\cdot F$ with a cyclic cubic field F   except for $k_4\cdot k_7^{+}$ and $k_4\cdot k_{3^2}^{+}$ of conductors 28 and 36.