Sums of integral squares in complex bi-quadratic fields and in CM fields

2021/03/09 by Srijonee Shabnam Chaudhury, Chaudhury, Srijonee Shabnam
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2103.05322

openalex publication_date 2021/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a complex bi-quadratic field with ring of integers O_K. For K = ℚ(√(-m), √(n)), where m ≡ 3 \pmod 4 and n ≡ 1 \pmod 4, we prove that every algebraic integer can be written as sum of integral squares. Using this, we prove that for any complex bi-quadratic field K, every element of 4O_K can be written as sum of five integral squares. In addition, we show that the Pythagoras number of ring of integers of any CM field is at most five. Moreover, we give two classes of complex bi-quadratic fields for which p(O_K)= 3 and p(4O_K)=3 respectively. Here, p(O_K) is the Pythagoras number of ring of integers of K and p(4O_K) is the smallest positive integer t such that every element of 4O_K can be written as sum of t integral squares.

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