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Articles 1 - 5 of 5
Full-Text Articles in Number Theory
Proportion Of P-Adic Polynomials Which Are Irreducible, Isaac Rajagopal
Proportion Of P-Adic Polynomials Which Are Irreducible, Isaac Rajagopal
Rose-Hulman Undergraduate Mathematics Journal
We attempt to quantify the exact proportion of p-adic polynomials of degree n which are irreducible. We find an exact answer to this when n is prime and p != n, and also when n = 4 and p != 2. Our answers are rational functions in p. This relates to previous work done to find exact proportions of p-adic polynomials of degree n which have k roots.
Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles
Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles
All Dissertations
This dissertation will explore factorization within orders in a number ring. By far the most well-understood of these orders are rings of algebraic integers. We will begin by examining how certain types of subrings may relate to the larger rings in which they are contained. We will then apply this knowledge, along with additional techniques, to determine how the elasticity in an order relates to the elasticity of the full ring of algebraic integers. Using many of the same strategies, we will develop a corresponding result in the rings of formal power series. Finally, we will explore a number of …
Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren
Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren
Senior Honors Theses
It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …
Algorithmic Factorization Of Polynomials Over Number Fields, Christian Schulz
Algorithmic Factorization Of Polynomials Over Number Fields, Christian Schulz
Mathematical Sciences Technical Reports (MSTR)
The problem of exact polynomial factorization, in other words expressing a polynomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used …
Commutative N-Ary Arithmetic, Aram Bingham
Commutative N-Ary Arithmetic, Aram Bingham
LSU New Orleans Theses and Dissertations
Motivated by primality and integer factorization, this thesis introduces generalizations of standard binary multiplication to commutative n-ary operations based upon geometric construction and representation. This class of operations are constructed to preserve commutativity and identity so that binary multiplication is included as a special case, in order to preserve relationships with ordinary multiplicative number theory. This leads to a study of their expression in terms of elementary symmetric polynomials, and connections are made to results from the theory of polyadic (n-ary) groups. Higher order operations yield wider factorization and representation possibilities which correspond to reductions in the set of primes …