On the strongly ambiguous classes of some biquadratic number fields

Abdelmalek Azizi; Abdelkader Zekhnini; Mohammed Taous

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 3, page 363-384
  • ISSN: 0862-7959

Abstract

top
We study the capitulation of 2 -ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields 𝕜 = ℚ ( 2 p q , i ) , where i = - 1 and p ≡ - q ≡ 1 ( mod 4 ) are different primes. For each of the three quadratic extensions 𝕂 / 𝕜 inside the absolute genus field 𝕜 ( * ) of 𝕜 , we determine a fundamental system of units and then compute the capitulation kernel of 𝕂 / 𝕜 . The generators of the groups Am s ( 𝕜 / F ) and Am ( 𝕜 / F ) are also determined from which we deduce that 𝕜 ( * ) is smaller than the relative genus field ( 𝕜 / ℚ ( i ) ) * . Then we prove that each strongly ambiguous class of 𝕜 / ℚ ( i ) capitulates already in 𝕜 ( * ) , which gives an example generalizing a theorem of Furuya (1977).

How to cite

top

Azizi, Abdelmalek, Zekhnini, Abdelkader, and Taous, Mohammed. "On the strongly ambiguous classes of some biquadratic number fields." Mathematica Bohemica 141.3 (2016): 363-384. <http://eudml.org/doc/286792>.

@article{Azizi2016,
abstract = {We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\mathbb \{k\}=\mathbb \{Q\}(\sqrt\{2pq\}, \{\rm i\})$, where $\{\rm i\}=\sqrt\{-1\}$ and $p\equiv -q\equiv 1 \hspace\{4.44443pt\}(\@mod \; 4)$ are different primes. For each of the three quadratic extensions $\mathbb \{K\}/\mathbb \{k\}$ inside the absolute genus field $\mathbb \{k\}^\{(*)\}$ of $\mathbb \{k\}$, we determine a fundamental system of units and then compute the capitulation kernel of $\mathbb \{K\}/\mathbb \{k\}$. The generators of the groups $\{\rm Am\}_s(\mathbb \{k\}/F)$ and $\{\rm Am\}(\mathbb \{k\}/F)$ are also determined from which we deduce that $\mathbb \{k\}^\{(*)\}$ is smaller than the relative genus field $(\mathbb \{k\}/\mathbb \{Q\}(\{\rm i\}))^*$. Then we prove that each strongly ambiguous class of $\mathbb \{k\}/\mathbb \{Q\}(\{\rm i\})$ capitulates already in $\mathbb \{k\}^\{(*)\}$, which gives an example generalizing a theorem of Furuya (1977).},
author = {Azizi, Abdelmalek, Zekhnini, Abdelkader, Taous, Mohammed},
journal = {Mathematica Bohemica},
keywords = {absolute genus field; relative genus field; fundamental system of units; 2-class group; capitulation; quadratic field; biquadratic field; multiquadratic CM-field},
language = {eng},
number = {3},
pages = {363-384},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the strongly ambiguous classes of some biquadratic number fields},
url = {http://eudml.org/doc/286792},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Azizi, Abdelmalek
AU - Zekhnini, Abdelkader
AU - Taous, Mohammed
TI - On the strongly ambiguous classes of some biquadratic number fields
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 3
SP - 363
EP - 384
AB - We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\mathbb {k}=\mathbb {Q}(\sqrt{2pq}, {\rm i})$, where ${\rm i}=\sqrt{-1}$ and $p\equiv -q\equiv 1 \hspace{4.44443pt}(\@mod \; 4)$ are different primes. For each of the three quadratic extensions $\mathbb {K}/\mathbb {k}$ inside the absolute genus field $\mathbb {k}^{(*)}$ of $\mathbb {k}$, we determine a fundamental system of units and then compute the capitulation kernel of $\mathbb {K}/\mathbb {k}$. The generators of the groups ${\rm Am}_s(\mathbb {k}/F)$ and ${\rm Am}(\mathbb {k}/F)$ are also determined from which we deduce that $\mathbb {k}^{(*)}$ is smaller than the relative genus field $(\mathbb {k}/\mathbb {Q}({\rm i}))^*$. Then we prove that each strongly ambiguous class of $\mathbb {k}/\mathbb {Q}({\rm i})$ capitulates already in $\mathbb {k}^{(*)}$, which gives an example generalizing a theorem of Furuya (1977).
LA - eng
KW - absolute genus field; relative genus field; fundamental system of units; 2-class group; capitulation; quadratic field; biquadratic field; multiquadratic CM-field
UR - http://eudml.org/doc/286792
ER -

References

top
  1. Azizi, A., On the capitulation of the 2-class group of 𝕜 = ℚ ( 2 p q , i ) where p ≡ - q ≡ 1 mod 4 , Acta Arith. 94 French (2000), 383-399. (2000) MR1779950
  2. Azizi, A., Units of certain imaginary abelian number fields over ℚ , Ann. Sci. Math. Qué. 23 French (1999), 15-21. (1999) Zbl1041.11072MR1721726
  3. Azizi, A., Taous, M., Determination of the fields 𝕜 = ℚ ( d , - 1 ) given the 2-class groups are of type ( 2 , 4 ) or ( 2 , 2 , 2 ) , Rend. Ist. Mat. Univ. Trieste 40 French (2008), 93-116. (2008) MR2583453
  4. Azizi, A., Zekhnini, A., Taous, M., On the strongly ambiguous classes of 𝕜 / ℚ ( i ) where 𝕜 = ℚ ( 2 p 1 p 2 , i ) , Asian-Eur. J. Math. 7 (2014), Article ID 1450021, 26 pages. (2014) MR3189588
  5. Azizi, A., Zekhnini, A., Taous, M., Structure of G a l ( 𝕜 2 ( 2 ) / 𝕜 ) for some fields 𝕜 = ℚ ( 2 p 1 p 2 , i ) with C l 2 ( 𝕜 ) ≃ ( 2 , 2 , 2 ) , Abh. Math. Semin. Univ. Hamb. 84 (2014), 203-231. (2014) MR3267742
  6. Azizi, A., Zekhnini, A., Taous, M., On the generators of the 2 -class group of the field 𝕜 = ℚ ( d , i ) , Int. J. of Pure and Applied Math. 81 (2012), 773-784. (2012) MR2974674
  7. Azizi, A., Zekhnini, A., Taous, M., On the unramified quadratic and biquadratic extensions of the field ℚ ( d , i ) , Int. J. Algebra 6 (2012), 1169-1173. (2012) Zbl1284.11142MR2974674
  8. Chevalley, C., Sur la théorie du corps de classes dans les corps finis et les corps locaux, J. Fac. Sci., Univ. Tokyo, Sect. (1) 2 French (1933), 365-476. (1933) Zbl0008.05301
  9. Furuya, H., 10.1016/0022-314X(77)90045-2, J. Number Theory 9 (1977), 4-15. (1977) Zbl0347.12006MR0429820DOI10.1016/0022-314X(77)90045-2
  10. Gras, G., Class Field Theory: From Theory to Practice, Springer Monographs in Mathematics Springer, Berlin (2003). (2003) Zbl1019.11032MR1941965
  11. Hasse, H., On the class number of abelian number fields, Mathematische Lehrbücher und Monographien I Akademie, Berlin German (1952). (1952) Zbl0046.26003MR0834791
  12. Heider, F.-P., Schmithals, B., Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. Reine Angew. Math. 336 German (1982), 1-25. (1982) Zbl0505.12016MR0671319
  13. Hirabayashi, M., Yoshino, K.-I., 10.1016/0022-314X(90)90141-D, J. Number Theory 34 (1990), 346-361. (1990) Zbl0705.11065MR1049510DOI10.1016/0022-314X(90)90141-D
  14. Kubota, T., 10.1017/S0027763000000088, Nagoya Math. J. 10 German (1956), 65-85. (1956) Zbl0074.03001MR0083009DOI10.1017/S0027763000000088
  15. Lemmermeyer, F., The ambiguous class number formula revisited, J. Ramanujan Math. Soc. 28 (2013), 415-421. (2013) MR3158989
  16. Louboutin, S., 10.1002/1522-2616(200007)215:1<107::AID-MANA107>3.0.CO;2-A, Math. Nachr. 215 (2000), 107-113. (2000) Zbl0972.11105MR1768197DOI10.1002/1522-2616(200007)215:1<107::AID-MANA107>3.0.CO;2-A
  17. McCall, T. M., Parry, C. J., Ranalli, R., 10.1006/jnth.1995.1079, J. Number Theory 53 (1995), 88-99. (1995) Zbl0831.11059MR1344833DOI10.1006/jnth.1995.1079
  18. Sime, P. J., 10.1090/S0002-9947-1995-1333398-3, Trans. Am. Math. Soc. 347 (1995), 4855-4876. (1995) Zbl0847.11060MR1333398DOI10.1090/S0002-9947-1995-1333398-3
  19. Terada, F., 10.2748/tmj/1178242555, Tohoku Math. J. (2) 23 (1971), 697-718. (1971) Zbl0243.12003MR0306158DOI10.2748/tmj/1178242555
  20. Wada, H., On the class number and the unit group of certain algebraic number fields, J. Fac. Sci., Univ. Tokyo, Sect. (1) 13 (1966), 201-209. (1966) Zbl0158.30103MR0214565

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.