Marques, Sophie; Ward, Kenneth An explicit triangular integral basis for any separable cubic extension of a function field. (English) Zbl 1437.11155 Eur. J. Math. 5, No. 4, 1252-1266 (2019). Let \({\mathbb F}_q(x)\) be a rational congruence function field of characteristic \(p\). Let \(L\) be a separable cubic extension of \({\mathbb F}_q(x)\) and let \({\mathcal O}_{L,x}\) denote the integral closure of \({\mathbb F}_q[x]\) in \(L\). An integral basis \(\mathfrak{B}\) of \({\mathcal O}_{L,x}/{\mathbb F}_q [x]\) is called triangular if there exists \(\omega\in{\mathcal O}_{L,x}\) such that the field base change matrix from \(\{1, \omega,\omega^2\}\) to \(\mathfrak{B}\) is upper triangular. In order to construct \(\mathfrak{B}\), the authors find the determinant of the base change matrix from an integral basis of \(L/{\mathbb F}_q(x)\) to \(\{1,\omega,\omega^2\}\). Next it is found from \(\{1,\omega,\omega^2\}\) a transformation to an integral ideal \(\mathfrak{B}\) with discriminant equal to that of \(L/{\mathbb F}_q(x)\). Thus the generators of \(\mathfrak{B}\) form an integral basis.In a previous work [Eur. J. Math. 5, No. 2, 551–570 (2019; Zbl 1429.11193)], the authors proved that any separable cubic extension has a generator with minimal polynomial either \(X^3-a\) (purely cubic) or \(X^3-3X-a\) when \(p\neq 3\) and \(X^3+aX+a^2\) when \(p=3\) (impurely cubic). In this paper, it is determined explicitly a triangular integral basis for impurely cubic extensions: Theorem 3.1 for characteristic different from \(3\) and Theorem 4.1 for characteristic \(3\). The case of purely cubic extensions was described by R. Scheidler in [J. Théor. Nombres Bordx. 13, No. 2, 609–631 (2001; Zbl 0995.11064)].In Section 2, the authors compute explicitly the discriminant for any separable impurely cubic function field over \({\mathbb F}_q(x)\) and in Section 3 those computations are used to determine that a triangular integral basis always exists and it is found such a basis explicitly for any characteristic. Reviewer: Gabriel D. Villa Salvador (Ciudad de México) Cited in 2 Documents MSC: 11R58 Arithmetic theory of algebraic function fields 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 11R16 Cubic and quartic extensions 11T55 Arithmetic theory of polynomial rings over finite fields Keywords:integral bases; cubic extensions; function fields; separable extensions; finite fields; global function fields; congruence function fields Citations:Zbl 1429.11193; Zbl 0995.11064 PDFBibTeX XMLCite \textit{S. Marques} and \textit{K. Ward}, Eur. J. Math. 5, No. 4, 1252--1266 (2019; Zbl 1437.11155) Full Text: DOI arXiv References: [1] Cassels, J.W.S., Fröhlich, A. (eds.): Algebraic Number Theory. Thompson, Washington (1967) · Zbl 0153.07403 [2] Conrad, K.: Galois groups of cubics and quartics in all characteristics (2013). http://www.math.uconn.edu/ kconrad/blurbs/galoistheory/ [3] Hasse, H.: Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172, 37-54 (1935) · Zbl 0010.00501 [4] Landquist, E., Rozenhart, P., Scheidler, R., Webster, J., Wu, Q.: An explicit treatment of cubic function fields with applications. Canadian J. Math. 62(4), 787-807 (2010) · Zbl 1213.14053 [5] Madan, M.L., Madden, D.J.: The exponent of class groups in congruence function fields. Acta Arith. 32(2), 183-205 (1977) · Zbl 0371.12010 [6] Marques, S., Ward, K.: A complete classification of cubic function fields over any finite field (2016). arXiv:1612.03534 [7] Marques, S., Ward, K.: Cubic fields: a primer (2017). arXiv:1703.06219 · Zbl 1429.11193 [8] Marques, S., Ward, K.: Cubic fields: a primer (2017, accepted) · Zbl 1429.11193 [9] Scheidler, R.: Ideal arithmetic and infrastructure in purely cubic function fields. J. Théor. Nombres Bordeaux 13(2), 609-631 (2001) · Zbl 0995.11064 [10] Scheidler, R.; Buell, D. (ed.), Algorithmic aspects of cubic function fields, No. 3076, 395-410 (2004), Berlin · Zbl 1125.11363 [11] Villa Salvador, G.D.: Topics in the Theory of Algebraic Function Fields. Mathematics: Theory and Applications. Birkhäuser, Boston (2006) · Zbl 1154.11001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.