Pure Quintic Fields Defined by Trinomials

Decomposition of primes in number fields defined by trinomials

On Quadratic Fields Generated by Discriminants of Irreducible Trinomials

A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant ∆n(a, b) of the trinomial fn,a,b(t) = t n + at+ b, where n ≥ 5 is a fixed integer. In particular, it is shown that, under the abc-conjecture, for every n ≡ 1 (mod 4), the quadratic fields Q (√ ∆n(a, b) ) are pairwise distinct for a positive proportion of such discriminants with i...

Roots of Trinomials over Prime Fields

The origin of this work was the search for a “Descartes’ rule” for finite fields a nontrivial upper bound for the number of roots of sparse polynomials. In [2], Bi, Cheng, and Rojas establish such an upper bound. Then, in [3], Cheng, Gao, Rojas, and Wan show that the bound is essentially optimal in an infinite number of cases by constructing t-nomials with many roots in Fpt . However, the bound...

Extremal Trinomials over Quadratic Finite Fields

In the process of pursuing a finite field analogue of Descartes’ Rule, Bi, Cheng, and Rojas (2014) proved an upper bound of 2 √ q − 1 on the number of roots of a trinomial c1 + c2x a2 + c3x a3 ∈ Fq [x], conditional on the exponents satisfying δ = gcd(a2, a3, q − 1) = 1, and Cheng, Gao, Rojas, and Wan (2015) showed that this bound is near-optimal for many cases. Our project set out to refine the...

Quadratic extensions of totally real quintic fields

In this work, we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than 1013 in absolute value. For each field in the list we give its discriminant, the discriminant of its subfield, a relative polynomial generating the field over one of its subfields, the corresponding polynomial over Q, and the Galois group of ...