On the norms of units in quadratic fields

Trotter, H. F.

doi:10.1090/s0002-9939-1969-0244196-6

Cited by 11 publications

(10 citation statements)

References 2 publications

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“…Since q r = −1, by a result of Dirichlet (cf. [14]), we get that the fundamental unit of K 2 = Q( √ qr) has norm −1. Therefore, by Lemma 2, we have a 2 = 1.…”

Section: Proof Of Theoremmentioning

confidence: 93%

“…A result of Dirichlet (cf. [14]) implies that for two distinct prime numbers r and s, if the Legendre symbol r s = −1, then the fundamental unit of Q( √ rs) has norm −1. Thus the hypotheses of Theorem 1, Theorem 2 and Theorem 3 imply that the fundamental units of the subfields Q( √ qr) and Q( √ pq) of K ′ , F and K ′′ respectively, have norm −1.…”

Section: Statements Of Main Theoremsmentioning

confidence: 99%

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Biquadratic fields having a non-principal Euclidean ideal class

Chattopadhyay

Muthukrishnan

2019

Journal of Number Theory

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H. W. Lenstra [7] introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in [3] and [6]). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves [3] and C. Hsu [6].2010 Mathematics Subject Classification. 11A05.

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“…Since q r = −1, by a result of Dirichlet (cf. [14]), we get that the fundamental unit of K 2 = Q( √ qr) has norm −1. Therefore, by Lemma 2, we have a 2 = 1.…”

Section: Proof Of Theoremmentioning

confidence: 93%

Section: Statements Of Main Theoremsmentioning

confidence: 99%

Biquadratic fields having a non-principal Euclidean ideal class

Chattopadhyay

Muthukrishnan

2019

Journal of Number Theory

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“…More precisely, we will give lower bounds for The first result in this direction goes back to Redei [1936;1939]. 1989;Hurrelbrink 1990;Pumpliin 1968;Trotter 1969]. …”

Section: Proven Densities For the Negative Pell Equationmentioning

confidence: 99%

The Number of Real Quadratic Fields Having Units of Negative Norm

Stevenhagen¹

1993

Experimental Mathematics

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“…. , P n ) to give a sufficient condition for N ( 0 ) = g. The proof is an analogue of the proof in [14] for quadratic number fields. Proof.…”

Section: Definition 22 (See Bae and Koomentioning

confidence: 83%

Capitulation problem for global function fields

Li¹,

2011

Arch. Math.

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Let q be a power of an odd prime number p, k = Fq(t) be the rational function field over the finite field Fq. In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields. Mathematics Subject Classification (2010). Primary 11R58;Secondary 11A15, 11R37.

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On the norms of units in quadratic fields

Cited by 11 publications

References 2 publications

Biquadratic fields having a non-principal Euclidean ideal class

Biquadratic fields having a non-principal Euclidean ideal class

The Number of Real Quadratic Fields Having Units of Negative Norm

Capitulation problem for global function fields

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