Ideal structure of polynomial rings over finite fields
*Jay Prakash TiwariCorresponding authordrjayprakashtiwari31@gmail.comDepartment of MathematicsPatel Group of InstitutionsIndore, Madhya Pradesh, 452020, IndiaView full profile → , D. Balajibalaji@avit.ac.inDepartment of Humanities and Sciences (Mathematics Division)Aarupadai Veedu Institute of TechnologyVinayaka Mission’s Research Foundation (Deemed to be University)Paiyanoor, Chennai, Tamil Nadu, 603104, IndiaView full profile → , Swagat M. Karveswagatkarve@gmail.comDepartment of Electronics & Telecommunication EngineeringS B Patil College of EngineeringIndapur, Maharashtra, 413106, IndiaView full profile → , Mridul Dixitmridul.dixit@gla.ac.inDepartment of Computer Engineering & ApplicationsGLA UniversityMathura, Uttar Pradesh, 281406, IndiaView full profile → , Preeti Pandurang Kalepreetikale2025@gmail.comDepartment of Electronics & Computer EngineeringCSMSS Chh. Shahu College of EngineeringChhatrapati Sambhajinagar, Maharashtra, 431011, IndiaView full profile → , Baru Prashanthiprashanthi1800@vardhaman.orgDepartment of Computer Science and EngineeringVardhaman College of EngineeringShamshabad, Hyderabad, Telangana, 501218, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 31 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-3031
- Pages:
- 4055–4064
Abstract
The paper discusses the best form of the polynomial rings on the finite fields with a consideration on their algebraic properties and their applications in the modern day algebra. With the assumption that F is a finite field, it is possible to generate ideals and analyze systematically the irreducible polynomials by the unique factorization of the ring F[x]. The two main concepts, maximal ideals and the description of quotient rings F[x]/(f(x)) are discussed and it is observed that they are connected with the field extensions. Special consideration is paid to the role of irreducibility in the establishment of field structures and homomorphic mappings. Theoretical results demonstrate that the perfect decomposition simplifies the factorization of polynomials, and helps to compute effectively in an algebraic system. Its results assist in gaining a better insight into structural behavior in finite algebraic structures and its application in coding theory, cryptography, and computational algebraic applications.
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References
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