Prime and Primitive Ideals of Ultragraph Leavitt Path Algebras [PDF]
arXiv, 2021 Let $\mathcal G$ be an ultragraph and let $K$ be a field. We describe prime and primitive ideals in the ultragraph Leavitt path algebra $L_K(\mathcal G)$. We identify the graded prime ideals in terms of downward directed sets and then we characterize the non-graded prime ideals.
arxiv
Solvable extensions of number fields ramified at only one prime are Ostrowski [PDF]
arXiv, 2023 In this note, we show that, under a certain condition, solvable extensions of number fields ramifed at only one prime are Ostrowski. As a corollary, we deduce a generalization of Hilbert Theorem 94 to cyclic extensions ramifed at one prime.
arxiv
An optical Eratosthenes' sieve for large prime numbers [PDF]
Optics Express 28, 11965-11973 (2020), 2019 We report the first experimental demonstration of prime number sieve via linear optics. The prime numbers distribution is encoded in the intensity zeros of the far field produced by a spatial light modulator hologram, which comprises a set of diffraction gratings whose periods correspond to all prime numbers below 149. To overcome the limited far field
arxiv +1 more source
Distribution of primes represented by polynomials and Multiple Dedekind zeta functions [PDF]
arXiv, 2022 n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider
arxiv
Explicit Bound for the Prime Ideal Theorem in Residue Classes [PDF]
arXiv, 2017 We give explicit numerical estimates for the generalized Chebyshev functions. Explicit results of this kind are useful for estimating of computational complexity of algorithms which generates special primes. Such primes are needed to construct an elliptic curve over prime field using complex multiplication method.
arxiv
Number-theoretic expressions obtained through analogy between prime factorization and optical interferometry [PDF]
, 2011 Prime factorization is an outstanding problem in arithmetic, with important consequences in a variety of fields, most notably cryptography. Here we employ the intriguing analogy between prime factorization and optical interferometry in order to obtain, for the first time, analytic expressions for closely related functions, including the number of ...
arxiv +1 more source
Explicit Factorization of Prime Integers in Quartic Number Fields defined by $X^4+aX+b$ [PDF]
arXiv, 2010 For every prime integer $p$, an explicit factorization of the principal ideal $p\z_K$ into prime ideals of $\z_K$ is given, where $K$ is a quartic number field defined by an irreducible polynomial $X^4+aX+b\in\z[X]$.
arxiv
Sets of Completely Decomposed Primes in Extensions of Number Fields [PDF]
arXiv, 2015 We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.
arxiv
On actions of connected Hopf algebras [PDF]
arXiv, 2020 Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained in $I$ is also prime (semi-prime, completely prime).
arxiv
Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields [PDF]
arXiv, 2014 In this paper, we introduce a Carlitz module analogue of Mersenne primes, and prove Carlitz module analogues of several classical results concerning Mersenne primes. In contrast to the classical case, we can show that there are infinitely many composite Mersenne numbers.
arxiv