A characterization of C-fields via Galois groups

Mináč, Ján; Smith, Tara L.

doi:10.1016/0021-8693(91)90078-m

Cited by 13 publications

(10 citation statements)

References 5 publications

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“…These results onZV, Q%-and DC-extensions can all be found in [M&S,Corollary 1.3], with exactly the same proof.…”

Section: Ii-»p1supporting

confidence: 67%

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On 2-Groups as Galois Groups

Ledet¹

1995

Can. j. math.

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Let L/K be a finite Galois extension in characteristic ≠ 2, and consider a non-split Galois theoretical embedding problem over L/K with cyclic kernel of order 2. In this paper, we prove that if the Galois group of L/K is the direct product of two subgroups, the obstruction to solving the embedding problem can be expressed as the product of the obstructions to related embedding problems over the corresponding subextensions of L/K and certain quaternion algebra factors in the Brauer group of K. In connection with this, the obstructions to realising non-abelian groups of order 8 and 16 as Galois groups over fields of characteristic ≠ 2 are calculated, and these obstructions are used to consider automatic realisations between groups of order 4, 8 and 16.

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“…These results onZV, Q%-and DC-extensions can all be found in [M&S,Corollary 1.3], with exactly the same proof.…”

Section: Ii-»p1supporting

confidence: 67%

“…By the result in Section 3, we have quadratically independent elements a,b,c E K*, such that (a, b) - (c, c). By [M&S,Proposition A. 1 ] there exists an x G K*, such that (a,bx) = (c,cx) = (ac,x) = 1.…”

Section: Ii-»p1mentioning

confidence: 99%

On 2-Groups as Galois Groups

Ledet¹

1995

Can. j. math.

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“…The same system of equations was given by M. Horie [13]; moreover, the construction presented by Minác and Smith [28] uses three equations which can easily be shown to be equivalent to (I)-(III). In the form of the "common slot property" (cf.…”

Section: Construction Of H 8 -Extensionsmentioning

confidence: 96%

Unramified quaternion extensions of quadratic number fields

Lemmermeyer¹

1997

Journal De Théorie Des Nombres De Bordeaux

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“…Taking √ a = sw, √ b = sy, √ c = s, and d = disc(q 1 (x)) in [27], Appendix, we have disc(q 1 (x))disc(q 2 (x)) = k a , disc(q 1 (x))disc(q 3 (x)) = k c , and disc(q 1 (x))disc(q 4 (x)) = k ac . Using Eqs.…”

Section: Fundamental Identities Whenmentioning

confidence: 99%

HT90 and “simplest” number fields

Foster¹

2011

Illinois J. Math.

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A standard formula (1) leads to a proof of HT90, but requires proving the existence of θ such that α = 0, so that β = α/σ(α).We instead impose the condition (M), that taking θ = 1 makes α = 0. Taking n = 3, we recover Shanks's simplest cubic fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic quartic fields, and a certain family of totally real cyclic extensions of Q(cos(π/4m)) all have defining polynomials whose zeroes satisfy (M).Further investigation of (M) for n = 4 leads to an elementary algebraic construction of a 2-parameter family of octic polynomials with "generic" Galois group 8 T 11 . Imposing an additional algebraic condition on these octics produces a new family of cyclic quartic extensions. This family includes the "simplest" quartic fields and Washington's cyclic quartic fields as special cases.We obtain more detailed results on our octics when the parameters are algebraic integers in a number field. In particular, we identify certain sets of special units, including exceptional sequences of 3 units, and give some of their properties.

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A characterization of C-fields via Galois groups

Cited by 13 publications

References 5 publications

On 2-Groups as Galois Groups

On 2-Groups as Galois Groups

Unramified quaternion extensions of quadratic number fields

HT90 and “simplest” number fields

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