On coefficient valuations of Eisenstein polynomials

Totally Ramified Primes and Eisenstein Polynomials

is Eisenstein at a prime p when each coefficient ci is divisible by p and the constant term c0 is not divisible by p 2. Such polynomials are irreducible in Q[T ], and this Eisenstein criterion for irreducibility is the way nearly everyone first meets Eisenstein polynomials. Here, we will show Eisenstein polynomials are closely related to total ramification of primes in number fields. Let K be a...

Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations

EEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations

GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...

T. N. Shorey’s Influence in the Theory of Irreducible Polynomials

The idea of looking at the prime factorization of the coefficients of a polynomial in Z[x] in order to establish its irreducibility (over Q) goes back to the classical Schönemann-Eisenstein criterion first derived in [29] and [6] in the middle of the 19th century. At the beginning of the 20th century, G. Dumas [5], again making use of primes that divide the coefficients of a polynomial, introdu...

Harmonic theta series and equidistribution

is a holomorphic modular form of weight 4 for the subgroup [1] Γθ, from the fact that there are no weight 4 cuspforms for Γθ, and from explicit computation of the Fourier coefficients of the two types of weight 4 Eisenstein series for Γθ. Toward a less frivolous result, recall that the irreducibility of spaces Hd of homogeneous, degree d harmonic polynomials f on R as O(n)-spaces gives Hecke’s ...