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An explicit triangular integral basis for any separable cubic extension of a function field. (English) Zbl 1437.11155

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.

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
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References:

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