Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 21 (2026), 177 -- 209
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THE STRUCTURE OF THE n-TH ROOTS OF UNITY IN RESIDUE RINGS OF PRIME IDEALS P OVER p IN ALGEBRAIC NUMBER FIELDS

Boaz Cohen

Abstract. Let 𝕂 be an algebraic number field and let O𝕂 be its ring of integers. In this paper, we study the structure of incongruent solutions of xn ≡ 1 (mod Pa) in O𝕂, where P is a prime ideal, in order to apply these results to solving xn = 1 over the P-adic field 𝕂P.

2020 Mathematics Subject Classification: 11-02; 11Y40; 11R04; 11K41; 11F85
Keywords: congruences; p-adic field; algebraic fields; roots of unity

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References

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Boaz Cohen, ORCID: https://orcid.org/0009-0004-7641-4522
Department of Computer Science,
The Academic College of Tel-Aviv,
Rabenu Yeruham St., P.O.B 8401 Yaffo, 6818211, Israel.
email: arctanx@gmail.com


Received: November 14, 2025; Accepted: April 23, 2026
Published electronically: April 23, 2026