{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,27]],"date-time":"2026-04-27T03:48:02Z","timestamp":1777261682246,"version":"3.51.4"},"reference-count":152,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T00:00:00Z","timestamp":1719360000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled \u201cMonogenity and Power Integral Bases\u201d. We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given.<\/jats:p>","DOI":"10.3390\/axioms13070429","type":"journal-article","created":{"date-parts":[[2024,6,26]],"date-time":"2024-06-26T09:29:33Z","timestamp":1719394173000},"page":"429","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Monogenity and Power Integral Bases: Recent Developments"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8266-9570","authenticated-orcid":false,"given":"Istv\u00e1n","family":"Ga\u00e1l","sequence":"first","affiliation":[{"name":"Mathematical Institute, University of Debrecen, Pf.400, H-4002 Debrecen, Hungary"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,26]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"\u00dcber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der h\u00f6heren Kongruenzen","volume":"23","author":"Dedekind","year":"1878","journal-title":"G\u00f6ttingen Abh."},{"key":"ref_2","unstructured":"Hensel, K. (1908). Theorie der Algebraischen Zahlen, Teubner."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hasse, H. (1963). Zahlentheorie, Akademie.","DOI":"10.1515\/9783112478202"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ga\u00e1l, I. (2019). Diophantine equations and power integral bases. Theory and algorithms, Birkh\u00e4user. [2nd ed.].","DOI":"10.1007\/978-3-030-23865-0"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1112\/plms\/s3-24.3.385","article-title":"Finiteness theorems for binary forms with given discriminant","volume":"24","author":"Birch","year":"1972","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_6","first-page":"141","article-title":"Sur les polyn\u00f4mes a coefficients entiers et de discriminant donne, III","volume":"23","year":"1976","journal-title":"Publ. Math. (Debrecen)"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Neukirch, J. (1999). Algebraic Number Theory, Springer.","DOI":"10.1007\/978-3-662-03983-0"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Cohen, H. (1993). A Course in Computational Algebraic Number Theory, GTM 138, Springer.","DOI":"10.1007\/978-3-662-02945-9"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Pohst, M.P., and Zassenhaus, H. (1989). Algorithmic Algebraic Number Theory, Campridge University Press. Encyclopedia of Mathematics and Its Applications.","DOI":"10.1017\/CBO9780511661952"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1090\/S0002-9947-1930-1501535-0","article-title":"On the common index divisor of an algebraic number field","volume":"32","author":"Engstrom","year":"1930","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1007\/BF01459087","article-title":"Newtonsche Polygone in der Theorie der algebraischen Korper","volume":"99","author":"Ore","year":"1928","journal-title":"Math. Ann."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/0021-8693(92)90071-S","article-title":"On a theorem of Ore","volume":"146","author":"Montes","year":"1992","journal-title":"J. Algebra"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1250073","DOI":"10.1142\/S0219498812500739","article-title":"Newton polygons and p-integral bases of quartic number fields","volume":"11","author":"Fadil","year":"2012","journal-title":"J. Algebra Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2050188","DOI":"10.1142\/S0219498820501881","article-title":"On Newton polygon techniques and factorization of polynomial over Henselian valued fields","volume":"19","author":"Fadil","year":"2020","journal-title":"J. Algebra Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1090\/S0002-9947-2011-05442-5","article-title":"Newton polygons of higher order in algebraic number theory","volume":"364","author":"Guardia","year":"2012","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_16","first-page":"61","article-title":"Integral bases and monogenity of pure number fields with non-square free parameters up to degree 9","volume":"83","author":"Fadil","year":"2023","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma Algebra System. I. The User Language","volume":"24","author":"Bosma","year":"1997","journal-title":"J. Symb. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1006\/jsco.1993.1064","article-title":"On the resolution of index form equations in quartic number fields","volume":"16","author":"Pohst","year":"1993","journal-title":"J. Symb. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1006\/jnth.1996.0035","article-title":"Simultaneous representation of integers by a pair of ternary quadratic forms\u2014With an application to index form equations in quartic number fields","volume":"57","author":"Pohst","year":"1996","journal-title":"J. Number Theory"},{"key":"ref_20","unstructured":"Mordell, L.J. (1969). Diophantine Equations, Academic Press."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2257","DOI":"10.1142\/S1793042114500778","article-title":"Power integral bases for certain pure sextic fields","volume":"10","author":"Ahmad","year":"2014","journal-title":"Int. J. Number Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3","DOI":"10.51286\/albjm\/1495919797","article-title":"A note on the monogeneity of power maps","volume":"11","author":"Gassert","year":"2017","journal-title":"Albanian J. Math."},{"key":"ref_23","unstructured":"Fadil, L.E. (2021). A note on monogenity of pure number fields. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.jnt.2020.09.004","article-title":"Monogenic pure cubics","volume":"219","author":"Aygin","year":"2021","journal-title":"J. Number Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"219","DOI":"10.5486\/PMD.2022.9138","article-title":"On power integral bases for certain pure number fields","volume":"100","author":"Fadil","year":"2022","journal-title":"Publ. Math. Debr."},{"key":"ref_26","first-page":"11","article-title":"On power integral bases for certain pure number fields defined by x18 \u2212 m","volume":"63","author":"Fadil","year":"2022","journal-title":"Commentat. Math. Univ. Carol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1007\/s40863-021-00254-z","article-title":"On monogenity of certain pure number fields defined by x20 \u2212 m","volume":"16","author":"Fadil","year":"2022","journal-title":"Sao Paulo J. Math. Sci."},{"key":"ref_28","first-page":"397","article-title":"On power integral bases for certain pure number fields defined by x24 \u2212 m","volume":"57","author":"Fadil","year":"2020","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_29","first-page":"371","article-title":"On power integral bases for certain pure number fields defined by x36 \u2212 m","volume":"58","author":"Fadil","year":"2021","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s40590-021-00388-2","article-title":"On power integral bases of certain pure number fields defined by x42 \u2212 m","volume":"27","author":"Fadil","year":"2021","journal-title":"Bol. Soc. Mat. Mex. III. Ser."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s40306-022-00481-2","article-title":"On monogenity of certain pure number fields defined by x60 \u2212 m","volume":"48","author":"Fadil","year":"2023","journal-title":"Acta Math. Vietnam"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"197","DOI":"10.33044\/revuma.2836","article-title":"On power integral bases of certain pure number fields defined by x84 \u2212 m","volume":"65","author":"Fadil","year":"2023","journal-title":"Rev. Uni\u00f3n Mat. Argent."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1007\/s40863-021-00251-2","article-title":"On power integral bases of certain pure number fields defined by x3r \u2212 m","volume":"16","author":"Yakkou","year":"2022","journal-title":"Sao Paulo J. Math. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"269","DOI":"10.4064\/aa210402-8-5","article-title":"On power integral bases for certain pure number fields defined by x2\u00b73k \u2212 m","volume":"201","author":"Fadil","year":"2021","journal-title":"Acta Arith."},{"key":"ref_35","first-page":"143","article-title":"On power integral bases for certain pure sextic fields","volume":"40","author":"Fadil","year":"2022","journal-title":"Bol. Soc. Parana. Mat."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2916","DOI":"10.1080\/00927872.2021.1883642","article-title":"On power integral bases for certain pure number fields defined by x2r\u00b75s \u2212 m","volume":"49","author":"Yakkou","year":"2021","journal-title":"Commun. Algebra"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"307","DOI":"10.4064\/cm8574-6-2021","article-title":"On power integral bases of certain pure number fields defined by x3r\u00b77s \u2212 m","volume":"169","author":"Fadil","year":"2022","journal-title":"Colloq. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/s44146-022-00039-6","article-title":"On monogenity of certain pure number fields defined by x2u\u00b73v \u2212 m","volume":"88","author":"Fadil","year":"2022","journal-title":"Acta Sci. Math."},{"key":"ref_39","first-page":"138","article-title":"On monogenity of certain pure number fields defined by x2r\u00b77s \u2212 m","volume":"41","author":"Fadil","year":"2023","journal-title":"Bol. Soc. Parana. Mat."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Fadil, L.E. (2022). On monogenity of certain pure number fields defined by x2u\u00b73v\u00b75t \u2212 m. arXiv.","DOI":"10.1007\/s44146-022-00039-6"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2235","DOI":"10.1142\/S1793042121500858","article-title":"On monogenity of certain pure number fields defined by xpr \u2212 m","volume":"17","author":"Yakkou","year":"2021","journal-title":"Int. J. Number Theory"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.jnt.2021.03.025","article-title":"On integral bases and monogeneity of pure sextic number fields with non-squarefree coefficients","volume":"228","author":"Fadil","year":"2021","journal-title":"J. Number Theory"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.jnt.2016.09.009","article-title":"Integral bases and monogenity of pure fields","volume":"173","author":"Remete","year":"2017","journal-title":"J. Number Theory"},{"key":"ref_44","unstructured":"Fadil, L.E., and Ga\u00e1l, I. (2022). On integral bases and monogenity of pure octic number fields with non-square free parameters. arXiv."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1080\/00927872.2023.2262041","article-title":"A note on indices of quartic number fields defined by trinomials x4 + ax + b","volume":"52","author":"Fadil","year":"2024","journal-title":"Commun. Algebra"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Fadil, L.E., and Ga\u00e1l, I. (2022). On the monogenity of quartic number fields defined by x4 + ax2 + b. arXiv.","DOI":"10.21203\/rs.3.rs-2486778\/v1"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"257","DOI":"10.4064\/aa180423-24-8","article-title":"Two families of monogenic S4 quartic number fields","volume":"186","author":"Smith","year":"2018","journal-title":"Acta Arith."},{"key":"ref_48","unstructured":"Jones, L. (2024). Monogenic cyclic quartic trinomials. arXiv."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1017\/S0013091522000529","article-title":"Common index divisor of the number fields defined by x5 + ax + b","volume":"65","author":"Jakhar","year":"2022","journal-title":"Proc. Edinb. Math. Soc. II. Ser."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"3102","DOI":"10.1080\/00927872.2022.2025820","article-title":"On common index divisors and monogenity of certain number fields defined by x5 + ax2 + b","volume":"50","author":"Fadil","year":"2022","journal-title":"Commun. Algebra"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"2355","DOI":"10.2989\/16073606.2022.2156000","article-title":"On the index divisors and monogenity of number fields defined by x5 + ax3 + b","volume":"46","author":"Fadil","year":"2023","journal-title":"Quaest. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1609","DOI":"10.2989\/16073606.2022.2110537","article-title":"On common index divisor and monogenity of certain number fields defined by trinomials X6 + AX + B","volume":"46","author":"Fadil","year":"2023","journal-title":"Quaest. Math."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"788","DOI":"10.4153\/S0008439521000825","article-title":"On nonmonogenic number fields defined by x6 + ax + b","volume":"65","author":"Jakhar","year":"2022","journal-title":"Can. Math. Bull."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.jnt.2021.10.017","article-title":"On non monogenity of certain number fields defined by trinomials x6 + ax3 + b","volume":"239","author":"Fadil","year":"2022","journal-title":"J. Number Theory"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Fadil, L.E., and Kchit, O. (2022). On index divisors and monogenity of certain sextic number fields defined by x6 + ax5 + b. arXiv.","DOI":"10.21203\/rs.3.rs-2486778\/v1"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"833","DOI":"10.2989\/16073606.2022.2043948","article-title":"A note on non-monogenity of number fields arised from sextic trinomials","volume":"46","author":"Jakhar","year":"2023","journal-title":"Quaest. Math."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"299","DOI":"10.2140\/involve.2022.15.299","article-title":"Monogenic fields arising from trinomials","volume":"15","author":"Ibarra","year":"2022","journal-title":"Involve"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2349","DOI":"10.1080\/00927872.2022.2159035","article-title":"On index divisors and monogenity of certain septic number fields defined by x7 + ax3 + b","volume":"51","author":"Fadil","year":"2023","journal-title":"Commun. Algebra"},{"key":"ref_59","unstructured":"Yakkou, H.B. (2022). The index of the septic number field defined by x7 + ax5 + b. arXiv."},{"key":"ref_60","unstructured":"Jakhar, A., Kaur, S., and Kumar, S. (2023). On common index divisor of the number fields defined by x7 + ax + b. arXiv."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1007\/s10474-022-01206-5","article-title":"On monogenity of certain number fields defined by x8 + ax + b","volume":"166","author":"Yakkou","year":"2022","journal-title":"Acta Math. Hung."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s10474-023-01353-3","article-title":"On the index of the octic number field defined by x8 + ax + b","volume":"170","author":"Yakkou","year":"2023","journal-title":"Acta Math. Hung."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"145","DOI":"10.4064\/cm8799-3-2022","article-title":"Non-monogenity of certain octic number fields defined by trinomials","volume":"171","author":"Jakhar","year":"2023","journal-title":"Colloq. Math."},{"key":"ref_64","unstructured":"Jones, L. (2024). Monogenic even octic polynomials and their Galois groups. arXiv."},{"key":"ref_65","unstructured":"Kchit, O. (2023). On the index divisors and monogenity of certain nonic number fields. arXiv."},{"key":"ref_66","unstructured":"Yakkou, H.B., and Tiebekabe, P. (2022). On common index divisors and monogenity of of the nonic number field defined by a trinomial x9 + ax + b. arXiv."},{"key":"ref_67","unstructured":"Fadil, L.E., and Kchit, O. (2023). The index of certain nonic number fields defined by x9 + ax2 + b. arXiv."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/s11139-023-00768-4","article-title":"On index divisors and monogenity of certain number fields defined by x12 + axm + b","volume":"63","author":"Fadil","year":"2024","journal-title":"Ramanujan J."},{"key":"ref_69","doi-asserted-by":"crossref","unstructured":"Yakkou, H.B. (2022). On monogenity of certain number fields defined by x2r + axm + b. arXiv.","DOI":"10.7169\/facm\/1987"},{"key":"ref_70","first-page":"199","article-title":"On monogenity of certain number fields defined by trinomials","volume":"67","author":"Yakkou","year":"2022","journal-title":"Funct. Approx. Comment. Math."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"2450095","DOI":"10.1142\/S0219498824500956","article-title":"Non-monogenity of some number fields generated by binomials or trinomials of prime-power degree","volume":"23","author":"Jakhar","year":"2024","journal-title":"J. Algebra Appl."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1216\/rmj.2023.53.103","article-title":"On nonmonogenic algebraic number fields","volume":"53","author":"Jakhar","year":"2023","journal-title":"Rocky Mt. J. Math."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.1142\/S1793042116501384","article-title":"On power basis of a class of algebraic number fields","volume":"12","author":"Jhorar","year":"2016","journal-title":"Int. J. Number Theory"},{"key":"ref_74","first-page":"685","article-title":"On nonmonogenic number fields defined by trinomials of type xn + axm + b","volume":"53","author":"Yakkou","year":"2023","journal-title":"Rocky Mt. J. Math."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1515\/ms-2023-0063","article-title":"On index and monogenity of certain number fields defined by trinomials","volume":"73","author":"Fadil","year":"2023","journal-title":"Math. Slovaca"},{"key":"ref_76","first-page":"650","article-title":"Nonmonogenity of number fields defined by trinomials","volume":"28","author":"Jakhar","year":"2022","journal-title":"N. Y. J. Math."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.jnt.2016.02.021","article-title":"On prime divisors of the index of an algebraic integer","volume":"166","author":"Jakhar","year":"2016","journal-title":"J. Number Theory"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"2505","DOI":"10.1142\/S1793042117501391","article-title":"Characterization of primes dividing the index of a trinomial","volume":"13","author":"Jakhar","year":"2017","journal-title":"Int. J. Number Theory"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"2250026","DOI":"10.1142\/S0219498822500268","article-title":"Monogenic reciprocal trinomials and their Galois groups","volume":"21","author":"Jones","year":"2022","journal-title":"J. Algebra Appl."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1007\/s11139-020-00310-w","article-title":"Sextic reciprocal monogenic dihedral polynomials","volume":"56","author":"Jones","year":"2021","journal-title":"Ramanujan J."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"95","DOI":"10.14232\/actasm-021-463-3","article-title":"Infinite families of non-monogenic trinomials","volume":"87","author":"Jones","year":"2021","journal-title":"Acta Sci. Math."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"1850039","DOI":"10.1142\/S0129167X18500398","article-title":"Infinite families of monogenic trinomials and their Galois groups","volume":"29","author":"Jones","year":"2018","journal-title":"Int. J. Math."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"2150089","DOI":"10.1142\/S0129167X21500890","article-title":"Monogenic trinomials with non-squarefree discriminant","volume":"32","author":"Jones","year":"2021","journal-title":"Int. J. Math."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.jnt.2018.09.026","article-title":"A family of monogenic S4 quartic fields arising from elliptic curves","volume":"197","author":"Gassert","year":"2019","journal-title":"J. Number Theory"},{"key":"ref_85","unstructured":"Yakkou, H.B. (2024). On indices and monogenity of quartic number fields defined by quadrinomials. arXiv."},{"key":"ref_86","doi-asserted-by":"crossref","unstructured":"Harrington, J., and Jones, L. (2024). Monogenic quartic polynomials and their Galois groups. arXiv.","DOI":"10.1017\/S000497272400073X"},{"key":"ref_87","unstructured":"Jakhar, A., and Kalwaniya, R. (2023). On the index divisors of certain number fields. arXiv."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1090\/proc\/14858","article-title":"Monogenic polynomials with non-squarefree discriminant","volume":"148","author":"Jones","year":"2020","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1515\/ms-2022-0039","article-title":"On necessary and sufficient conditions for the monogeneity of a certain class of polynomials","volume":"72","author":"Jones","year":"2022","journal-title":"Math. Slovaca"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/s00009-023-02522-y","article-title":"On power basis of a class of number fields","volume":"20","author":"Jakhar","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"2448","DOI":"10.1080\/00927872.2022.2162913","article-title":"On non-monogenity of the number fields defined by certain quadrinomials","volume":"51","author":"Jakhar","year":"2023","journal-title":"Commun. Algebra"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"2117","DOI":"10.1216\/rmj.2020.50.2117","article-title":"On primes dividing the index of a quadrinomial","volume":"50","author":"Jakhar","year":"2020","journal-title":"Rocky Mt. J. Math."},{"key":"ref_93","first-page":"1465","article-title":"Infinite families of reciprocal monogenic polynomials and their Galois groups","volume":"27","author":"Jones","year":"2021","journal-title":"New York J. Math."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"213","DOI":"10.4064\/aa200211-21-7","article-title":"Some new infinite families of monogenic polynomials with non-squarefree discriminant","volume":"197","author":"Jones","year":"2021","journal-title":"Acta Arith."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1017\/S0004972719000182","article-title":"A brief note on some infinite families of monogenic polynomials","volume":"100","author":"Jones","year":"2019","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"37","DOI":"10.51286\/albjm\/1608313765","article-title":"Generating infinite families of monogenic polynomials using a new discriminant formula","volume":"14","author":"Jones","year":"2020","journal-title":"Albanian J. Math."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1017\/S0004972722000193","article-title":"0 Reciprocal monogenic quintinomials of degree 2n","volume":"106","author":"Jones","year":"2022","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4064\/cm8552-4-2021","article-title":"Infinite families of monogenic quadrinomials, quintinomials and sextinomials","volume":"169","author":"Jones","year":"2022","journal-title":"Colloq. Math."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1016\/j.jnt.2012.04.006","article-title":"Generating a power basis over a Dedekind ring","volume":"132","author":"Charkani","year":"2012","journal-title":"J. Number Theory"},{"key":"ref_100","first-page":"367","article-title":"A note on generating a power basis over a Dedekind ring","volume":"58","author":"Deajim","year":"2021","journal-title":"Stud. Sci. Math. Hung."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"117","DOI":"10.21136\/MB.2022.0142-21","article-title":"On relative pure cyclic fields with power integral bases","volume":"148","author":"Sahmoudi","year":"2023","journal-title":"Math. Bohem."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1216\/rmj.2021.51.1443","article-title":"On relative power integral basis of a family of numbers fields","volume":"51","author":"Soullami","year":"2021","journal-title":"Rocky Mt. J. Math."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s40590-022-00472-1","article-title":"On relative monogeneity of a family of number fields defined by Xpn + aXps \u2212 b","volume":"29","author":"Boughaleb","year":"2023","journal-title":"Bol. Soc. Mat. Mex., III. Ser."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"313","DOI":"10.4064\/aa200811-7-10","article-title":"The monogeneity of radical extensions","volume":"198","author":"Smith","year":"2021","journal-title":"Acta Arith."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"1650091","DOI":"10.1142\/S0219498816500912","article-title":"When is R[\u03d1] integrally closed?","volume":"15","author":"Khanduja","year":"2016","journal-title":"J. Algebra Appl."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1007\/s40993-022-00419-5","article-title":"The scheme of monogenic generators I: Representability","volume":"9","author":"Arpin","year":"2023","journal-title":"Res. Number Theory"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s40993-023-00449-7","article-title":"The scheme of monogenic generators. II: Local monogenicity and twists","volume":"9","author":"Arpin","year":"2023","journal-title":"Res. Number Theory"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.jnt.2020.12.006","article-title":"Rikuna\u2019s generic cyclic polynomial and the monogenity","volume":"231","author":"Sekigawa","year":"2022","journal-title":"J. Number Theory"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.11650\/tjm\/200201","article-title":"Monogenic binomial compositions","volume":"24","author":"Harrington","year":"2020","journal-title":"Taiwan. J. Math."},{"key":"ref_110","unstructured":"Jakhar, A., Kalwaniya, R., and Yadav, P. (2024). A study of monogenity of binomial composition. arXiv."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"115","DOI":"10.2996\/kmj44107","article-title":"Monogenic cyclotomic compositions","volume":"44","author":"Harrington","year":"2021","journal-title":"Kodai Math. J."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"85","DOI":"10.51286\/albjm\/1635847634","article-title":"Monogenically stable polynomials","volume":"15","author":"Jones","year":"2021","journal-title":"Albanian J. Math."},{"key":"ref_113","first-page":"93","article-title":"The monogenity of power-compositional Eisenstein polynomials","volume":"55","author":"Jones","year":"2022","journal-title":"Ann. Math. Inform."},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"93","DOI":"10.7169\/facm\/2104","article-title":"On the monogenicity of power-compositional Shanks polynomials","volume":"69","author":"Jones","year":"2023","journal-title":"Funct. Approx. Comment. Math."},{"key":"ref_115","doi-asserted-by":"crossref","unstructured":"Jones, L. (2023). The monogenicity of power-compositional characteristic polynomials. arXiv.","DOI":"10.7169\/facm\/2104"},{"key":"ref_116","unstructured":"Harrington, J., and Jones, L. (2022). The Irreducibility and Monogenicity of Power-Compositional Trinomials. arXiv."},{"key":"ref_117","unstructured":"Kaur, S., Kumar, S., and Remete, L. (2024). On the index of power compositional polynomials. arXiv."},{"key":"ref_118","unstructured":"Jones, L. (2022). A connection between the monogenicity of certain power-compositional trinomials and k-Wall-Sun-Sun primes. arXiv."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"3","DOI":"10.51286\/albjm\/1678110273","article-title":"Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials","volume":"17","author":"Jones","year":"2023","journal-title":"Albanian J. Math."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"17","DOI":"10.11650\/tjm\/231003","article-title":"A new condition for k-Wall-Sun-Sun primes","volume":"28","author":"Jones","year":"2024","journal-title":"Taiwan. J. Math."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1017\/S0004972723000138","article-title":"A note on generalised Wall-Sun-Sun primes","volume":"108","author":"Harrington","year":"2023","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_122","unstructured":"Kang, M., and Kim, D. (2023). The proportion of monogenic orders of prime power indices of the pure cubic field. arXiv."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"307","DOI":"10.4064\/aa230120-16-7","article-title":"Orders with few rational monogenizations","volume":"210","author":"Evertse","year":"2023","journal-title":"Acta Arith."},{"key":"ref_124","doi-asserted-by":"crossref","unstructured":"Nathanson, M.B. (2020). Counting monogenic cubic orders. Combinatorial and Additive Number Theory III, Springer. Papers Based on Talks Given at the CANT 2017 and 2018 Workshops, New York, NY, USA, May 2017 and May 2018.","DOI":"10.1007\/978-3-030-31106-3"},{"key":"ref_125","unstructured":"Alp\u00f6ge, L., Bhargava, M., and Shnidman, A. (2020). A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so. arXiv."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"513","DOI":"10.5486\/PMD.2022.9433","article-title":"On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)","volume":"100","author":"Bhargava","year":"2022","journal-title":"Publ. Math. Debr."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"57","DOI":"10.2140\/ent.2022.1.57","article-title":"Quartic index form equations and monogenizations of quartic orders","volume":"1","author":"Akhtari","year":"2022","journal-title":"Essent. Number Theory"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/s11139-018-0049-0","article-title":"Monogenic dihedral quartic extensions","volume":"50","author":"Kim","year":"2019","journal-title":"Ramanujan J."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1016\/j.jnt.2015.06.018","article-title":"Monogenity of totally real algebraic extension fields over a cyclotomic field","volume":"158","author":"Khan","year":"2016","journal-title":"J. Number Theory"},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.jnt.2018.10.002","article-title":"An ideal theoretic proof on monogenity of cyclic sextic fields of prime power conductor","volume":"198","author":"Khan","year":"2019","journal-title":"J. Number Theory"},{"key":"ref_131","unstructured":"Gil-Munoz, D., and Tinkov\u00e1, M. (2022). Additive structure of non-monogenic simplest cubic fields. arXiv."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"233","DOI":"10.11650\/tjm\/211003","article-title":"Monogenic Pisot and anti-Pisot polynomials","volume":"26","author":"Jones","year":"2022","journal-title":"Taiwan J. Math."},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"106281","DOI":"10.1016\/j.jpaa.2019.106281","article-title":"On the index of an algebraic integer and beyond","volume":"224","author":"Jakhar","year":"2020","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"806","DOI":"10.4153\/S0008439521000874","article-title":"A dynamical characterization for monogenity at every level of some infinite 2-towers","volume":"65","author":"Castillo","year":"2022","journal-title":"Canad. Math. Bull."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"290","DOI":"10.2996\/kmj44204","article-title":"The characterization of cyclic cubic fields with power integral bases","volume":"44","author":"Kashio","year":"2021","journal-title":"Kodai Math. J."},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1007\/s13370-016-0476-2","article-title":"Chatelain\u2019s integer bases for biquadratic fields","volume":"28","year":"2017","journal-title":"Afr. Mat."},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s13370-016-0428-x","article-title":"Chatelain\u2019s integral bases for triquadratic number fields","volume":"28","author":"Kouakou","year":"2017","journal-title":"Afr. Mat."},{"key":"ref_138","first-page":"9","article-title":"Diophantine proof of non-monogeneity for triquadratic number fields with odd discriminant","volume":"15","author":"Kouakou","year":"2021","journal-title":"Fundam. J. Math. Math. Sci."},{"key":"ref_139","first-page":"117","article-title":"Generators of power integral bases of Q(\u03b624) = Q(\u22123, 2, \u22121)","volume":"5","author":"Kouassi","year":"2025","journal-title":"Ann. Math\u00e9matiques Afr."},{"key":"ref_140","unstructured":"Aruna, C., and Vanchinathan, P. (2023). Exceptional Quartics are Ubiquitous. arXiv."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"293","DOI":"10.5486\/PMD.2018.7848","article-title":"Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields","volume":"92","year":"2018","journal-title":"Publ. Math. Debr."},{"key":"ref_142","first-page":"87","article-title":"Monogenity in totally complex sextic fields, revisited","volume":"47","year":"2020","journal-title":"JP J. Algebra Number Theory Appl."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s44146-023-00081-y","article-title":"Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields","volume":"89","year":"2023","journal-title":"Acta Sci. Math."},{"key":"ref_144","first-page":"97","article-title":"An experiment on the monogenity of a family of trinomials","volume":"51","year":"2021","journal-title":"JP J. Algebra Number Theory Appl."},{"key":"ref_145","first-page":"1","article-title":"On the monogenity of certain binomial compositions","volume":"57","year":"2022","journal-title":"JP J. Algebra Number Theory Appl."},{"key":"ref_146","first-page":"85","article-title":"Calculating generators of power integral bases in pure sextic fields","volume":"70","year":"2024","journal-title":"Funct. Approx. Comment. Math."},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s44146-023-00092-9","article-title":"On the monogenity of pure quartic relative extensions of Q(i)","volume":"89","author":"Remete","year":"2023","journal-title":"Acta Sci. Math."},{"key":"ref_148","first-page":"85","article-title":"A note on the monogenity of totally complex pure sextic fields","volume":"60","year":"2023","journal-title":"JP J. Algebra Number Theory Appl."},{"key":"ref_149","doi-asserted-by":"crossref","unstructured":"Ga\u00e1l, I. (2024). On the monogenity of totally complex pure octic fields. arXiv.","DOI":"10.17654\/0972555523006"},{"key":"ref_150","first-page":"265","article-title":"A note on the monogenity of some trinomials of type x4 + ax2 + b","volume":"63","year":"2024","journal-title":"JP J. Algebra Number Theory Appl."},{"key":"ref_151","doi-asserted-by":"crossref","unstructured":"Ga\u00e1l, I. (2024). Calculating power integral bases in some quartic fields corresponding to monogenic families of polynomials. arXiv.","DOI":"10.7169\/facm\/2111"},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1006\/jnth.2000.2541","article-title":"Computing power integral bases in quartic relative extensions","volume":"85","author":"Pohst","year":"2000","journal-title":"J. Number Theory"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/429\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:04:42Z","timestamp":1760108682000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/429"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,26]]},"references-count":152,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070429"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070429","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,26]]}}}