| Title:
|
Sums of two integer squares in a certain quartic extension (English) |
| Author:
|
Zinevičius, Albertas |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2026 |
| Pages:
|
69-73 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
An example of a quartic extension of the rational number field that does not have quadratic subfields and there exists a set of rational prime numbers $p \equiv 3 \pmod{4}$ of positive Dirichlet density such that either $p$ or $31p$ is a sum of two squares of integers of the extension is given. (English) |
| Keyword:
|
Sums of two integer squares |
| Keyword:
|
quartic number field |
| MSC:
|
11D09 |
| MSC:
|
11R37 |
| DOI:
|
10.5817/AM2026-2-69 |
| . |
| Date available:
|
2026-06-03T08:16:39Z |
| Last updated:
|
2026-06-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153667 |
| . |
| Reference:
|
[1] : Sums of two squares in (certain) integral domains.https://mathoverflow.net/questions/30998/, 2010, Last accessed 6 May, 2026. |
| Reference:
|
[2] Childress, N.: Class Field Theory.Springer, 2009. |
| Reference:
|
[3] Janusz, G. J.: Algebraic number fields.Academic Press, 1973. |
| Reference:
|
[4] Milne, J. S.: Algebraic Number Theory.Version 3.03, jmilne.org, 2011. |
| Reference:
|
[5] Milne, J. S.: Fields and Galois Theory.Version 4.60, jmilne.org, 2018. |
| Reference:
|
[6] Murty, R., Esmonde, J.: Problems in Algebraic Number Theory.2 ed., Springer, 2005. |
| Reference:
|
[7] Nagell, T.: On the sum of two integral squares in certain quadratic fields.Ark. Mat. 4 (1961), 267–286. 10.1007/BF02592013 |
| Reference:
|
[8] Neukirch, J.: Algebraic Number Theory.Springer-Verlag Berlin Heidelberg, 1999. Zbl 0956.11021 |
| Reference:
|
[9] Niven, I.: Integers of quadratic fields as sums of squares.Trans. Amer. Math. Soc. 48 (1940), 405–417. 10.1090/S0002-9947-1940-0003000-5 |
| Reference:
|
[10] SageMath,: Version 9.2.Online, 2020, Available at: https://www.sagemath.org/. |
| Reference:
|
[11] Zinevičius, A.: Non-sums of two cubes of algebraic integers.Colloq. Math. 163 (2021), 285–293. 10.4064/cm7945-11-2019 |
| . |
