A generalization of simplest number fields and their integral basis
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A GENERALIZATION OF SIMPLEST NUMBER FIELDS AND THEIR INTEGRAL BASIS L. REMETE Mathematical Institute, University of Debrecen, Pf. 400, H-4002 Debrecen, Hungary e-mail: [email protected] (Received March 15, 2020; revised June 16, 2020; accepted June 22, 2020)
Abstract. An integral basis of the simplest number fields of degrees 3, 4 and 6 over Q is well-known, and widely investigated. We generalize the simplest number fields to any degree, and show that an integral basis of these fields is repeating periodically.
1. Introduction Let a, b, c, d ∈ Q and σ : C → C,
σ : z →
az + b . cz + d
Assume that f (X) ∈ Z[X] is a polynomial with real roots such that σ transitively permutes the roots of f (X). In this case, if β is a root of f (X), then Q(β) is a totally real cyclic number field. This requires that the matrix a b ∈ P GL2 (Q), M= c d is of finite order. It can be shown that each non-trivial torsion element of P GL2 (Q) has order 2, 3, 4 or 6, whence, there exist such polynomials and number fields only of degrees 2, 3, 4 and 6. According to D. Shanks [27], A.J. Lazarus [18], G. Lettl, A. Peth˝ o and P. Voutier [21] and A. Hoshi [14] the polynomials of degrees 3, 4 and 6 with these properties are called simplest polynomials, and the corresponding number fields are called simplest number fields. These fields have an extensive literature. ´ Supported through the UNKP-19-3 New National Excellence Program of the Ministry for Innovation and Technology. Key words and phrases: simplest polynomial, simplest number field, integral basis. Mathematics Subject Classification: 11C08, 11R04, 11R09, 11R21, 11R32.
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L. REMETE
First, the simplest cubic fields were investigated by H. Cohn [1] and D.Shanks [27], because they have easily computable and relatively large class numbers. M. N. Gras [9], [10], V. Ennola [2], [3] and A. J. Lazarus [18] investigated the unit group of the simplest fields. K. Foster [4] obtained the simplest parametric polynomials, just by using a special identity of units in cyclic extensions. It shows, that these fields have some other unique and interesting properties. E. Thomas [28], M. Mignotte [22], G. Lettl, A. Peth˝o and P. Voutier [21], [20] and G. Lettl and A. Peth˝o [19], I. Ga´al [5] solved Thue equations corresponding to the simplest polynomials in absolute case, and C.Heuberger [11], I. Ga´al, B. Jadrijevi´c and L. Remete [6] in certain relative cases. A. Hoshi [12], [13], [14] gave a correspondence between solutions of a family of Thue equations and the isomorphism classes of the simplest number fields. He extended his results to a family of polynomials of degree √ 12, which has similar properties as the simplest polynomials, but over Q( −3). This generalization provided the main motivation of our approach. Assume that β is a root of a simplest polynomial f (X) ∈ Z[X] of degree n, and the M¨obius transformation σ permutes its roots transitively. Then the conjugates of β are: {β, σ(β), σ 2 (β), . . . , σ n−1 (β)}.
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