Lenny Jones

Generalized Wieferich primes and monogenic polynomials

Lenny Jones

Let $b, p \in Z$ with $b \geq 2$ and $p \geq 3$ a prime. If $b^{p - 1} \equiv 1 (mod p^{2})$ , then $p$ is called a {\em generalized Wieferich prime base $b$ }, or more succinctly,…

Generalized Wieferich primes and monogenic trinomials

Amy Falk, Joshua Harrington, Lenny Jones

Let $b \geq 2$ be an integer and let $p \geq 3$ be a prime. We say that $p$ is a {\em generalized Wieferich prime base $b$ }, or more succinctly, a {\em base- $b$ Wieferich prime,} if $b…

A Note on Abelian Monogenic Trinomials

Lenny Jones

An abelian monogenic polynomial $f (x) \in Z [x]$ is a monic polynomial of degree $N$ that is irreducible over $Q$ , such that the Galois group of $f (x)$ over ${\ma…

Wieferich Primes and Monogenic Trinomials

Lenny Jones

A prime $p$ is called a Wieferich prime if $2^{p - 1} \equiv 1 (mod p^{2})$ . A monic polynomial $f (x) \in Z [x]$ of degree $N \geq 2$ is called monogenic if $f (x)$ is irreducible…

Characterizing monogenic trinomials $x^{12} + a x^{6} + b$ according to their Galois groups

Lenny Jones

Let $f (x) = x^{12} + a x^{6} + b \in Z [x]$ , with $ab \neq = 0$ . We say that $f (x)$ is {\em monogenic} if $f (x)$ is irreducible over $Q$ and ${1, θ, θ^{2}, \dots, θ^{11}}$ is a…

On the monogenicity and Galois groups of $x^{2 p} + a x^{p} + b^{p}$

Joshua Harrington, Lenny Jones

Let $f (x) = x^{2 p} + a x^{p} + b^{p}$ , where $p$ is a prime and $a, b \in Z$ with $ab \neq = 0$ . If $f (x)$ is irreducible over $Q$ , we say that $f (x)$ is monogenic if $\{1,θ,θ^2…

Monogenic even sextic trinomials and their Galois groups

Lenny Jones

Let $f (x) = x^{6} + A x^{2 k} + B \in Z [x]$ , with $A \neq = 0$ and $k \in {1, 2}$ . We say that $f (x)$ is {\em monogenic} if $f (x)$ is irreducible over $Q$ and $\{1,θ,θ^2,θ^3,θ…

Monogenic Strictly-Perron Polynomials

Lenny Jones

A monic polynomial $f (x) \in Z [x]$ of degree $n$ is called monogenic if $f (x)$ is irreducible over $Q$ and ${1, θ, θ^{2}, \dots, θ^{n - 1}}$ is a basis for the ring…

Monogenic sextic trinomials $x^{6} + A x^{3} + B$ and their Galois groups

Joshua Harrington, Lenny Jones

Let $f (x) = x^{6} + A x^{3} + B \in Z [x]$ , with $A \neq = 0$ , and suppose that $f (x)$ is irreducible over $Q$ . We define $f (x)$ to be {\em monogenic} if $\{1,θ,θ^2,θ^3,θ^4,θ^{5…

Monogenic Cyclic Polynomials in Recurrence Sequences

Joshua Harrington, Lenny Jones

Let $f (x) \in Z [x]$ be an $N$ th degree polynomial that is monic and irreducible over $Q$ . We say that $f (x)$ is {\em monogenic} if ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ …

Monogenic Reciprocal Quartic Polynomials And Their Galois Groups

Lenny Jones

Suppose that $f (x) = x^{4} + A x^{3} + B x^{2} + A x + 1 \in Z [x]$ . We say that $f (x)$ is monogenic if $f (x)$ is irreducible over $Q$ and ${1, θ, θ^{2}, θ^{3}}$ is a basis for the ring…

Monogenic Even Cyclic Sextic Polynomials

Lenny Jones

Suppose that $f (x) \in Z [x]$ is monic and irreducible over $Q$ of degree $N$ . We say that $f (x)$ is monogenic if ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ is a basis for the…

Monogenic trinomials and class numbers of related quadratic fields

Lenny Jones

We say that a monic polynomial $f (x) \in Z [x]$ of degree $N \geq 2$ is monogenic if $f (x)$ is irreducible over $Q$ and ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ is a basis for…

Monogenic cyclic trinomials of the form $x^{4} + c x + d$

Lenny Jones

A monic polynomial $f (x) \in Z [x]$ of degree $n$ that is irreducible over $Q$ is called cyclic if the Galois group over $Q$ of $f (x)$ is the cyclic gro…

The monogenicity and Galois groups of certain reciprocal quintinomials

Lenny Jones

We say that a monic polynomial $f (x) \in Z [x]$ of degree $N$ is monogenic if $f (x)$ is irreducible over $Q$ and ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ is a basis for ${\m…