Monogenity of totally real algebraic extension fields over a cyclotomic field

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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 Img" 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_n

img="si1.gif" overflow="scroll"> k_{n}

of odd conductor n鈮?

img="si2.gif" overflow="scroll"> n 鈮?/mo>3

or even one 鈮? with 4|n

img="si3.gif" overflow="scroll"> 4 | n

and a totally real algebraic extension field F over the rationals Q and both fields 15f97" title="Click to view the MathML source">k_n

img="si1.gif" overflow="scroll"> 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 15f97" title="Click to view the MathML source">k_n

img="si1.gif" overflow="scroll"> k_{n}

has no power integral basis. Each of the composite fields K is also a CM field over the maximal real subfield <img 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">

img="si4.gif" overflow="scroll"> k_{n}^{+} \cdot F

of K . This result involves the previous work for K=k_n⋅F

img="si5.gif" overflow="scroll"> K = k_{n} \cdot F

of the Eisenstein field k_n=k₃

img="si6.gif" overflow="scroll"> k_{n} = k_{3}

and the maximal real subfields <img 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">

img="si7.gif" overflow="scroll"> F = k_{p^{n}}^{+}

of prime power conductor pⁿ

img="si37.gif" overflow="scroll"> p^{n}

with p鈮?

img="si38.gif" overflow="scroll"> p 鈮?/mo>5

, and an analogue K=k_n⋅F

img="si5.gif" overflow="scroll"> K = k_{n} \cdot F

of cyclotomic fields <img 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">

img="si10.gif" overflow="scroll"> k_{n} = k_{2^{m}} (m 鈮?/mo>3)

with a totally real algebraic fields F of K=k₄⋅F

img="si11.gif" overflow="scroll"> K = k_{4} \cdot F

with a cyclic cubic field F except for <img 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">

img="si12.gif" overflow="scroll"> k_{4} \cdot k_{7}^{+}

and <img 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">

img="si13.gif" overflow="scroll"> k_{4} \cdot k_{3^{2}}^{+}

of conductors 28 and 36.

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