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On the number of incongruent solutions to a quadratic congruence over algebraic integers.
- Source :
-
International Journal of Number Theory . Feb2019, Vol. 15 Issue 1, p105-130. 26p. - Publication Year :
- 2019
-
Abstract
- Over the ring of algebraic integers 𝒪 of a number field K , the quadratic congruence a 1 x 1 2 + ⋯ + a k x k 2 ≡ c modulo a nonzero ideal ℐ is considered. We prove explicit formulas for N k (a 1 , ... , a k , c , ℐ) and N k ∗ (a 1 , ... , a k , c , ℐ) , the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k and the number of incongruent solutions (x 1 , ... , x k) ∈ 𝒪 k with x 1 x 2 ⋯ x k coprime to ℐ , respectively. If ℐ is contained in a prime ideal 𝒫 containing the rational prime p , it is assumed that 𝒫 is unramified over p. Moreover, some interesting identities for exponential sums are proved. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 17930421
- Volume :
- 15
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- International Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 134660508
- Full Text :
- https://doi.org/10.1142/S1793042118501762