Imaginary Bicyclic Biquadratic Fields With Cyclic 2-Class Group

McCall, Thomas Mark; Parry, Charles J.; Ranalli, Ramona Renee

doi:10.1006/jnth.1995.1079

Cited by 41 publications

(21 citation statements)

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“…where r = rankC k,2 = t − e − 1 and 2 e = [E F : E F ∩ N k/F (k × )] (see for example [20]). The relation between |Am(k/F )| and |Am s (k/F )| is given by the following formula (see for example [15]):…”

Section: Resultsmentioning

confidence: 99%

The Generators of the $2$-Class Group of Some Fields: Correction to Theorem 3 of }

Azizi¹,

Zekhnini²,

Taous³

2015

Int. J. of Pure and Appl. Math.

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Section: Resultsmentioning

confidence: 99%

The Generators of the $2$-Class Group of Some Fields: Correction to Theorem 3 of }

Azizi¹,

Zekhnini²,

Taous³

2015

Int. J. of Pure and Appl. Math.

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“…From Lemma 1 of [18] we know that (E : H ) = 1 if and only if d has no prime factors congruent to 3 mod 4 that split completely in F. We therefore see that d must contain exactly two or three prime factors. By the above criteria for (E:H), genus theory, the fact that (E:H) = 1 or 2 since F is imaginary quadratic, and the splitting of primes from Q to F corresponding to their Kronecker symbols, we see that the above conditions describe completely when rank Cl k,2 = 2.…”

Section: Imaginary Biquadratic Number Fields Withmentioning

confidence: 93%

On the 2-class field tower of some imaginary biquadratic number fields

Benjamin

2006

Ramanujan J

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be an imaginary biquadratic number field with Cl k,2 , the 2-class group of k, isomorphic to Z/2Z × Z/2 m Z, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1.

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“…Let us first calculate the 2-rank of Cl 2 (L 1 ). The number of the ramified odd primes in Q( √ p 2 q) is 2, among them there is only one which is congruent to 1 (mod 4); then, according to [17], the 2-rank of Cl 2 (L 1 ) is r 2 = 2 + 1 − 2 = 1; thus…”

Section: Generators Of the 2-class Groups Of Some Cm-fieldsmentioning

confidence: 98%

Coclass of ${\rm Gal}({\mathbb K}_2^{(2)}/{\mathbb K})$ for some fields ${\mathbb K} = {\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$ with 2-class groups of types (2, 2, 2)

2015

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Let p 1 ≡ p 2 ≡ −q ≡ 1 (mod 4) be primes such that ( 2 p 1 ) = 1 and ( 2 p 2 ) = ( p 1 p 2 ) = ( p 1 q ) = −1. Put i = √ −1 and d = p 1 p 2 q, then the bicyclic biquadratic field k = Q( √ d, i) has an elementary Abelian 2-class group of rank 3. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-Abelian Galois group Gal(k (2) 2 /k) of the second Hilbert 2-class field k (2) 2 of k, we study the 2-class field tower of k, and we study the capitulation problem of the 2-classes of k in its fourteen abelian unramified extensions of relative degrees two and four.

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Imaginary Bicyclic Biquadratic Fields With Cyclic 2-Class Group

Cited by 41 publications

References 0 publications

The Generators of the $2$-Class Group of Some Fields: Correction to Theorem 3 of }

The Generators of the $2$-Class Group of Some Fields: Correction to Theorem 3 of }

On the 2-class field tower of some imaginary biquadratic number fields

Coclass of ${\rm Gal}({\mathbb K}_2^{(2)}/{\mathbb K})$ for some fields ${\mathbb K} = {\mathbb Q}(\sqrt{p_1p_2q}, \sqrt{-1})$ with 2-class groups of types (2, 2, 2)

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