Galois Halkaları

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Thesis Type: Postgraduate

Institution Of The Thesis: Yıldız Technical University, Graduate School of Natural and Applied Sciences, MATEMATİK ANABİLİM DALI, Turkey

Approval Date: 2011

Thesis Language: Turkish

Student: Elif altınay

Principal Supervisor (For Co-Supervisor Theses): Gürsel Yeşilot

Open Archive Collection: AVESIS Open Access Collection

Abstract:

The main goal of this thesis is exploring the structure of Galois rings and the construction of them. First of all, in Section 2, we give information that is necessary for being able to study Galois rings and then chain rings. In Section 3, we study the residue class ring of , the ring of polinomials of , Hensel?s Lemma and basic irreducible polinomials. The information in this section has a crucial part in The Theory of Galois Rings. Fourth Section is about Galois Rings. In this section, we study the definition of Galois rings, their structure, p-adic represantation, units, extention and automorphisms, respectively and give some examples. Let p be a prime. In a finite commutative ring with identity, if the set of zero and zero-divisors form then this ring is called a Galois ring. The non-zero divisors are units in R, so is the unique maximal ideal of R. If is a monic basic irreducible then . Thus, any two Galois rings of the same characteristic and cardinality are isomorphic. Therefore, we can use the notation GR( , ) to denote any Galois ring of characteristic and cardinality . Further, there exists a a root of of order , and . Let and . Then, any element can be written uniquely as.Let and be two Galois rings. If is a monic basic irreducible polinomial of degree l in then. In the last section, we study chain rings. A ring is called a chain ring if all its ideals form a chain under inclusion. Finite chain rings are precisely finite local rings whose maximal ideal is principal. Let be a finite chain ring with maximal ideal M . Let be : the characteristic of ; : ; : the nilpotency index of M and such that where and where ,; k : the greatest integer that is equal or less than such that . Then the integers p, s, m, k, t are called as the invarianats of . Let be an Eisenstein polinomial of degree Then, any finite chain ring R whose invariants are p, s, m, k, t is .