Connection between Gaussian periods and cyclic units

Lehmer, Emma

doi:10.1090/s0025-5718-1988-0929551-0

Cited by 54 publications

(21 citation statements)

References 9 publications

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“…If m|b or m|c then property (6) follows from properties (3) and (4). Suppose that mAa, mAb, mAc, mAa + b and mAa + c. We have This proves property (5). This also proves property (6) when b ≡ c mod m. Suppose furthermore that mAa + b + c. If b ≡ c mod m, we have by (26)…”

Section: Propositionsupporting

confidence: 54%

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Cyclic polynomials and the multiplication matrices of their roots

Thaine

2004

Journal of Pure and Applied Algebra

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Let D be an integrally closed, characteristic zero domain, K its ÿeld of fractions, m ¿ 2 an integer and P(a cyclic polynomial. Let be a generator of Gal(K[Â0]=K) and suppose the Âi are labeled so that (Âi) = Âi+1 (indices mod m). Suppose that the discriminant discr K[Â 0 ]=K (Â0; Â1; : : : ; Âm−1) is nonzero. For 0 6 i; j 6 m−1, deÿne the elements ai;j ∈ K by Â0Âi = m−1 j=0 ai;jÂj. Let A = [ai;j] 06i; j¡m . We call A the multiplication matrix of the Âi. We have that P(x) is the characteristic polynomial of A. In this article we study the relations between P(x) and A. We show how to factor P(x) in the ÿeld K[A] and how to construct A in terms of the coe cients ci. We give methods to construct matrices A, with entries in K, such that the characteristic polynomial of A belongs to D[x], is cyclic and has A as the multiplication matrix of its roots. One of these methods derives from a natural composition of multiplication matrices. The other method gives matrices A that are generalizations of matrices of cyclotomic numbers of order m, whose characteristic polynomials have roots that are generalizations of (real and complex) Gaussian periods of degree m. As applications we construct families of polynomials with cyclic and dihedral Galois group over Q and with cyclic Galois groups over quadratic ÿelds.

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Section: Propositionsupporting

confidence: 54%

“…The ideas in Section 3 originated in a question posed by RenÃ e Schoof regarding a generalization of certain families of polynomials of degree 5 found by Emma Lehmer (see [5] and [7]). …”

Section: Multiplication Matricesmentioning

confidence: 99%

Cyclic polynomials and the multiplication matrices of their roots

Thaine

2004

Journal of Pure and Applied Algebra

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“…To construct quintic cyclic fields, we make use the polynomial f (T , X) introduced by Emma Lehmer [6] (see Section 1 for the definition). We will in fact construct quintic cyclic fields each having an unramified quadratic extension.…”

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confidence: 99%

A family of quintic cyclic fields with even class number parameterized by rational points on an elliptic curve

Nakano

2009

Journal of Number Theory

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“…Foster's paper [6] has an excellent review of earlier work on the Shanks polynomials and the simplest cubic fields; he also proved that every degree-three cyclic extension of the rationals is generated by a Shanks polynomial (which implies the same for our RSC); this was done earlier by Kersten and Michaliček [7]. Also, Lehmer [11] and Lazarus [10] have shown that the minimal polynomials for so-called cubic Gaussian periods, when composed with some x − a for a an integer, will equal one of the Shanks polynomials (and thus are related to our RSC's).…”

mentioning

confidence: 85%