Reverse Mathematics of Divisibility in Integral Domains

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Reverse Mathematics of Divisibility in Integral Domains

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Faktorisasi Polinomial Square-Free dan bukan Square-Free atas Lapangan Hingga Zp

Abstrak: Faktorisasi polinomial merupakan suatu proses penguraian suatu polinomial berderajat n menjadi polinomial-polinomial lain yang berderajat lebih kecil dari n. Faktorisasi polinomial atas lapangan hingga merupakan suatu proses pengerjaan yang relative tidak mudah. Oleh karena itu, diperlukan suatu metode yang berupa algoritma untuk memproses faktorisasi polinomial. Algoritma Faktorisasi Berlekamp merupakan salah satu metode terbaik dalam memfaktorisasi polinomial atas lapangan hingga . Polinomial atas lapangan terbagi dua kategori berdasarkan faktorisasinya, yaitu polinomial square-free dan bukan square-free. Polinomial square-free adalah polinomial dimana setiap faktorisasi tak tereduksi tunggal. Sedangkan bukan square-free adalah sebaliknya. Penelitian ini bertujuan untuk membuat suatu algoritma untuk menfaktorkan polinomial square-free dan bukan square-free atas lapangan hingga. Adapun (Divasὀn, Joosten, Thiemann, & Yamada, 2017) yang menjadi referensi utama dalam penelitian ini adalah berdasarkan. Namun, dibatasi hanya untuk polinomial square-free saja. Untuk itulah dengan menggunakan konsep polinomial faktorisasi ganda. Pada bagian akhir penelitian akan mengimplementasikan algoritma baru yang telah disusun. Abstract: Polynomial factorization is a decomposition of a polynomial of degree n into other polynomials whose degree is less than n. Polynomial factorization over finite field is a relatively easy in process. Therefore, it’s needed a method in the form of an algorithm to process polynomial factorization. Algorithm Factorization Berlekamp is one of the best methods in factoring polynomials over a finite field . Polynomials over field are divided into two category based on its factorization, namely square-free and not square-free polynomials. Square-free polynomials are polynomials in which each irreducible factorization is single. When non square-free is the opposite. This research aims to set an algorithm for factoring square-free polynomials and non square-free polynomials over a finite field . The main reference in this research is based on (Divasὀn, Joosten, Thiemann, & Yamada, 2017) (Saropah, 2012). However, it is restricted only to square-free polynomials. For this reason, this research will use the concept of repeated factorization polynomials. At the end of the research will implement a new algorithm that has been set.

On *-clean group rings

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

DIRECTED GRAPHS AND KRONECKER INVARIANTS OF PAIRS OF MATRICES

Call two pairs (M,N) and (M′,N′) of m × n matrices over a field K, simultaneously K-equivalent if there exist square invertible matrices S,T over K, with M′ = SMT and N′ = SNT. Kronecker [2] has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph Γ, with v vertices and e edges, an ordered pair (M,N) of e × v matrices of zeros and ones. It is natural to try to compute the Kronecker invariants of such a pair (M,N), particularly since they clearly furnish isomorphism-invariants of Γ. Let us call two graphs "linearly equivalent" when their two corresponding pairs are simultaneously equivalent. There have existed, since 1890, highly effective algorithms for computing the Kronecker invariants of pairs of matrices of the same size over a given field [1,2,5,6] and in particular for those arising in the manner just described from finite directed graphs. The purpose of the present paper, is to compute directly these Kronecker invariants of finite directed graphs, from elementary combinatorial properties of the graphs. A pleasant surprise is that these new invariants are purely rational — indeed, integral, in the sense that the computation needed to decide if two directed graphs are linearly equivalent only involves counting vertices in various finite graphs constructed from each of the given graphs — and does not involve finding the irreducible factorization of a polynomial over K (in apparent contrast both to the familiar invariant-computations of graphs furnished by the eigenvalues of the connection matrix, and to the isomorphism problem for general pairs of matrices).