Journal of Algebra and Its Applications最新文献

{"title":"Images of multilinear polynomials on generalized quaternion algebras","authors":"Peter V. Danchev, Truong Huu Dung, Tran Nam Son","doi":"10.1142/s0219498825502093","DOIUrl":"https://doi.org/10.1142/s0219498825502093","url":null,"abstract":"<p>In connection with the work of Malev published in [<i>J. Algebra Appl.</i><b>13</b> (2014) 1450004; <i>J. Algebra Appl.</i><b>20</b> (2021) 2150074], we continue to provide a classification of possible images of multilinear polynomials on generalized quaternion algebras.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"50 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Para-Kähler and pseudo-Kähler structures on Lie–Yamaguti algebras","authors":"Jia Zhao, Yuqin Feng, Yu Qiao","doi":"10.1142/s0219498825502044","DOIUrl":"https://doi.org/10.1142/s0219498825502044","url":null,"abstract":"<p>For a pre-Lie–Yamaguti algebra <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span>, by using its sub-adjacent Lie–Yamaguti algebra <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span><span></span>, we are able to construct a semidirect product Lie–Yamaguti algebra via a representation of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span><span></span>. The investigation of such semidirect Lie–Yamaguti algebras leads us to the notions of para-Kähler structures and pseudo-Kähler structures on Lie–Yamaguti algebras, and also gives the definition of complex product structures on Lie–Yamaguti algebras. Furthermore, a Levi–Civita product with respect to a pseudo-Riemannian Lie–Yamaguti algebra is introduced and we explore its relation with pre-Lie–Yamaguti algebras.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"49 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite p-groups in which the cores of all the nonnormal subgroups are in the center","authors":"Libo Zhao, Yangming Li, Lü Gong, Xiuyun Guo","doi":"10.1142/s0219498825502020","DOIUrl":"https://doi.org/10.1142/s0219498825502020","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> be a finite <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-group. Then <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> is said to be a <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mi>Z</mi></math></span><span></span>-group if <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>H</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>≤</mo><mi>Z</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> for every nonnormal subgroup <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>. In this paper, we study the <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>C</mi><mi>Z</mi></math></span><span></span>-group <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> and get <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>≤</mo><mn>3</mn></math></span><span></span>. It is proved that <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">exp</mtext></mstyle><mo stretchy=\"false\">(</mo><msup><mrow><mi>G</mi></mrow><mrow><mi>′</mi></mrow></msup><mo stretchy=\"false\">)</mo><mo>=</mo><mi>p</mi></math></span><span></span> if <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span><span></span> if <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><mi>c</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>3</mn></math></span><span></span>.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relative Gorenstein flat modules and Foxby classes and their model structures","authors":"Driss Bennis, Rachid El Maaouy, J. R. García Rozas, Luis Oyonarte","doi":"10.1142/s0219498825501944","DOIUrl":"https://doi.org/10.1142/s0219498825501944","url":null,"abstract":"<p>We introduce the concepts of relative (strongly) cotorsion and relative Gorenstein cotorsion modules for a non-necessarily semidualizing module and prove that there exists a unique hereditary abelian model structure where the cofibrations are the monomorphisms with relative Gorenstein flat cokernel and the fibrations are the epimorphisms with relative cotorsion kernel belonging to the Bass class. In the particular case of a semidualizing module, we investigate the existence of abelian model structures on the category of left (right) R-modules where the cofibrations are the epimorphisms (monomorphisms) with kernel (cokernel) belonging to the Bass (Auslander) class. We also show that the class of relative Gorenstein flat modules and the Bass class are part of weak AB-contexts.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"23 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the norm of the lower central series in a finite group","authors":"Lü Gong, Ziru Jing, Libo Zhao, Baojun Li","doi":"10.1142/s0219498825502032","DOIUrl":"https://doi.org/10.1142/s0219498825502032","url":null,"abstract":"<p>In this paper, the norm <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of the lower central series in a finite group <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> is introduced, which unifies the norm of derived subgroups and nilpotent residuals. Some propositions of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> are obtained, and some related subgroups as well as their equivalent propositions can also be found.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"145 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quandles with one nontrivial column","authors":"Nicholas Cazet","doi":"10.1142/s0219498825502019","DOIUrl":"https://doi.org/10.1142/s0219498825502019","url":null,"abstract":"<p>The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Hom</mtext></mstyle></math></span><span></span> quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cohomology and deformation theory of crossed homomorphisms of Leibniz algebras","authors":"Yizheng Li, Dingguo Wang","doi":"10.1142/s0219498825501956","DOIUrl":"https://doi.org/10.1142/s0219498825501956","url":null,"abstract":"<p>In this paper, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by crossed homomorphisms on Leibniz algebras. This allows us to define cohomology for a crossed homomorphism. Finally, we study linear deformations, formal deformations and extendibility of finite order deformations of a crossed homomorphism in terms of the cohomology theory.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Morita equivalences on Brauer algebras and BMW algebras of simply-laced types","authors":"Shoumin Liu","doi":"10.1142/s0219498825501749","DOIUrl":"https://doi.org/10.1142/s0219498825501749","url":null,"abstract":"<p>The Morita equivalences of classical Brauer algebras and classical Birman–Murakami–Wenzl (BMW) algebras have been well studied. Here, we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and BMW algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of Coxeter groups, respectively.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"110 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Certain linear isomorphisms for hyperalgebras relative to a Chevalley group","authors":"Yutaka Yoshii","doi":"10.1142/s0219498825501853","DOIUrl":"https://doi.org/10.1142/s0219498825501853","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> be a simply connected and simple algebraic group defined and split over a finite prime field <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span> of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span> elements. In this paper, using an <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span>-linear map splitting Frobenius endomorphism on a hyperalgebra relative to <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>, we obtain some <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>𝔽</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span><span></span>-linear isomorphisms induced by multiplication in the hyperalgebra.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The set of representatives and explicit factorization of xn − 1 over finite fields","authors":"Manjit Singh, Deepak","doi":"10.1142/s0219498825501701","DOIUrl":"https://doi.org/10.1142/s0219498825501701","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; be a positive integer and let &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;𝔽&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; be a finite field with &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; elements, where &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is a prime power and &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;gcd&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. In this paper, we give the explicit factorization of &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; over &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;𝔽&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and count the number of its irreducible factors for the following conditions: &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; are odd and &lt;span&gt;&lt;math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle&gt;&lt;mtext&gt;rad&lt;/mtext&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. First, we present a method to obtain the set of all representatives of &lt;span&gt;&lt;math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;-cyclotomic cosets modulo &lt;span&gt;&lt;math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;gcd&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. This set of representatives is then used to find the irreducible factors of &lt;span&gt;&lt;math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and the cyclotomic polynomial &lt;span&gt;&lt;math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"normal\"&gt;Φ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; over &lt;span&gt;&lt;math altimg=\"eq-00017.gif\" display=\"inline\" overflow=\"scroll","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"111 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}