The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m,… (More)

Let p be a prime number, K a number field, and S a finite set of places of K. Let KS be the compositum of all extensions of K (in a fixed algebraic closure K) which are unramified outside S, and put… (More)

Let K be a number field, and let λ(x, t) ∈ K[x, t] be irreducible over K(t). Using algebraic geometry and group theory, we study the set of α ∈ K for which the specialized polynomial λ(x, α) is… (More)

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R0(2m) be the minimal root discriminant for totally complex number fields of degree… (More)

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that L n (x) = Pn j=0 `n− j+r n− j ́ x / j!… (More)

Let F be the cubic field of discriminant −23 and let O ⊂ F be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL2(O), we computationally investigate modularity of… (More)

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α ∈ Q−Z<0, Filaseta and Lam have shown that the nth degree… (More)

Fix a prime number p, a number field K, and a finite set S of primes of K. Let Sp be the set of all primes of K of residue characteristic p. Inside a fixed algebraic closure K of K, let KS be the… (More)