Factoring octonion polynomials

We provide an analogue of Wedderburn’s factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree [Formula: see text] with [Formula: see text] conjugacy classes of roots, counting multiplicities.

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.

Factoring Polynomials Using Elliptic Curves

This paper presents a probabilistic algorithm to factor polynomials over finite fields using elliptic curves. The success of the algorithm depends on the initial choice of elliptic curve parameters. The algorithm is illustrated through numerical examples.

Hierarchies of Dignity

This chapter discusses what Rafael Bombelli called dignità, which translates to the English word “dignity” and is equivalent to what we refer to as “exponents.” It first considers Bombelli's L'Algebra (1579), which introduces a new kind of notation for the unknown and its powers. Written in Italian, L'Algebra used equality in a different sense than we do. For Bombelli, the higher powers meant hierarchies of dignity. He was not only inventing genuine symbols when depicting dignità as little cups holding numbers, but also inventing words that were new to mathematics. The chapter also examines how the problem of finding the roots of polynomials became the problem of factoring polynomials. Finally, it looks at René Descartes's idea of using numerical superscripts to mark positive integral exponents of a polynomial in his La Géométrie.