Note on the Hilbert 2-class field tower

Abdelmalek Azizi; Mohamed Mahmoud Chems-Eddin; Abdelkader Zekhnini

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 4, page 513-524
  • ISSN: 0862-7959

Abstract

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Let k be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields 𝕜 = ℚ ( d , - 1 ) , which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

How to cite

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Azizi, Abdelmalek, Chems-Eddin, Mohamed Mahmoud, and Zekhnini, Abdelkader. "Note on the Hilbert 2-class field tower." Mathematica Bohemica 147.4 (2022): 513-524. <http://eudml.org/doc/298797>.

@article{Azizi2022,
abstract = {Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $\mathbb \{k\}=\mathbb \{Q\}\big (\sqrt\{d\}, \sqrt\{-1\}\big )$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).},
author = {Azizi, Abdelmalek, Chems-Eddin, Mohamed Mahmoud, Zekhnini, Abdelkader},
journal = {Mathematica Bohemica},
keywords = {multiquadratic field; fundamental systems of units; 2-class group; 2-class field tower; capitulation},
language = {eng},
number = {4},
pages = {513-524},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Note on the Hilbert 2-class field tower},
url = {http://eudml.org/doc/298797},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Azizi, Abdelmalek
AU - Chems-Eddin, Mohamed Mahmoud
AU - Zekhnini, Abdelkader
TI - Note on the Hilbert 2-class field tower
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 4
SP - 513
EP - 524
AB - Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $\mathbb {k}=\mathbb {Q}\big (\sqrt{d}, \sqrt{-1}\big )$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).
LA - eng
KW - multiquadratic field; fundamental systems of units; 2-class group; 2-class field tower; capitulation
UR - http://eudml.org/doc/298797
ER -

References

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  1. Azizi, A., 10.1007/BF02844380, Rend. Circ. Mat. Palermo, II. Ser. French 48 (1999), 71-92. (1999) Zbl0920.11076MR1705171DOI10.1007/BF02844380
  2. Azizi, A., Unités de certains corps de nombres imaginaires et abéliens sur ℚ , Ann. Sci. Math. Qué. 23 (1999), 15-21 French. (1999) Zbl1041.11072MR1721726
  3. Azizi, A., Benhamza, I., Sur la capitulation des 2-classes d’idéaux de ℚ ( d , - 2 ) , Ann. Sci. Math. Qué. French 29 (2005), 1-20. (2005) Zbl1217.11097MR2296826
  4. Azizi, A., Zekhnini, A., 10.1142/S1793557120500539, Asian-Eur. J. Math. 13 (2020), Article ID 2050053, 6 pages. (2020) Zbl1442.11154MR4096421DOI10.1142/S1793557120500539
  5. Azizi, A., Zekhnini, A., Taous, M., 10.12732/ijpam.v103i1.8, Int. J. Pure Appl. Math. 103 (2015), 99-107. (2015) MR3405725DOI10.12732/ijpam.v103i1.8
  6. Chems-Eddin, M. M., Azizi, A., Zekhnini, A., 10.1007/s40590-021-00329-z, Bol. Soc. Mat. Mex., III. Ser. 27 (2021), Article ID 24, 16 pages. (2021) Zbl07342807MR4220815DOI10.1007/s40590-021-00329-z
  7. Chems-Eddin, M. M., Zekhnini, A., Azizi, A., 10.3906/mat-2003-117, Turk. J. Math. 44 (2020), 1466-1483. (2020) Zbl1455.11140MR4122918DOI10.3906/mat-2003-117
  8. Conner, P. E., Hurrelbrink, J., 10.1142/0663, Series in Pure Mathematics 8. World Scientific, Singapore (1988). (1988) Zbl0743.11061MR0963648DOI10.1142/0663
  9. Gorenstein, D., Finite Groups, Harper's Series in Modern Mathematics. Harper and Row, New York (1968). (1968) Zbl0185.05701MR0231903
  10. Heider, F.-P., Schmithals, B., 10.1515/crll.1982.336.1, J. Reine Angew. Math. 366 (1982), 1-25 German. (1982) Zbl0505.12016MR0671319DOI10.1515/crll.1982.336.1
  11. Hilbert, D., 10.1007/BF01446682, Math. Ann. 45 (1894), 309-340 German 9999JFM99999 25.0303.01. (1894) MR1510866DOI10.1007/BF01446682
  12. Kaplan, P., 10.1515/crll.1976.283-284.313, J. Reine Angew. Math. 283/284 (1976), 313-363 French. (1976) Zbl0337.12003MR0404206DOI10.1515/crll.1976.283-284.313
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  14. Wada, H., On the class number and the unit group of certain algebraic number fields, J. Fac. Sci., Univ. Tokyo, Sect. I 13 (1966), 201-209. (1966) Zbl0158.30103MR0214565

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