Managing Editor: Bruinier, Jan Hendrik
Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna
IMPACT FACTOR 2015: 0.823
Rank 88 out of 312 in category Mathematics and 124 out of 254 in Applied Mathematics in the 2015 Thomson Reuters Journal Citation Report/Science Edition
SCImago Journal Rank (SJR) 2015: 0.848
Source Normalized Impact per Paper (SNIP) 2015: 1.000
Impact per Publication (IPP) 2015: 0.606
Mathematical Citation Quotient (MCQ) 2015: 0.66
Embeddability of quadratic extensions in cyclic extensions
Citation Information: Forum Mathematicum. Volume 19, Issue 4, Pages 707–725, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/FORUM.2007.027, September 2007
- Published Online:
For an algebraic number field K we study the quadratic extensions of K which can be embedded in a cyclic extension of K of degree 2n for all natural numbers n, as well as the quadratic extensions which can be embedded in an infinite normal extension with the additive group of 2-adic integers as Galois group.
For shortness we call a normal extension of K whose Galois group is the cyclic group ℤ/2nℤ of order 2n with n ∈ ℕ, resp. , a (ℤ/2nℤ)-extension resp. a -extension of K.
A quadratic extension L|K is called (ℤ/2nℤ)-embeddable, resp. -embeddable, if there exists a (ℤ/2nℤ)-extension, resp. a -extension, of K containing L.
One main result of this paper is the following observation, the exact formulation of which is given in theorems 6 to 8 in §3:
Theorem 0. Let K be an imaginary quadratic number field whose discriminant has m prime divisors. Then the number of quadratic extensions L|K which are (ℤ/2nℤ)-embeddable for all n is 2m−1 − 1, 2m − 1 or 2m+1 − 1, depending on certain congruences for the discriminant and its prime divisors. But the number of quadratic extensions L|K which are -embeddable is only 3.
Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.