Narrow Results

Let $f(x)=x^4+a{x}^3+d\in \mathbb{Z}[x]$, where $ad\ne 0$. Let $C_n$ denote the cyclic group of order $n$, $D_4$ the dihedral group of order 8, and $A_4$ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only $a$ and $d$ to determine the Galois group $G$ of $f(x)$ over $\mathbb{Q}$. In particular, we show that $G=D_4$ if and only if $(a,d)=(\pm 2,2)$, and that $G\notin \{C_4,C_2\times C_2\}$. Furthermore, we prove that $f(x)$ is monogenic with $G=A_4$ if and only if $a=4k$ and $d=27k^4+1$, where $k\ne 0$ is an integer such that $27k^4+1$ is squarefree. This paper extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups.

Let $n\in \mathbb{Z}$ with $n\ge 3$. Let $S_n$ and $A_n$ denote, respectively, the symmetric group and alternating group on $n$ letters. Let $m$ be an indeterminate, and define

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+A{x}^m+B$ with non-squarefree discriminant, many of which have small Galois group. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.

An affine algebraic variety $X$ is rigid if the algebra of regular functions $𝕂[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the trinomial is at least $2$.

Consider an irreducible quartic polynomial of the form $X^4+aX+b$, where $a,b\in \mathbb{Z}$ satisfy $v_p(a)<3$ or $v_p(b)<4$, where $v_p$ denotes the exact power of a rational prime $p$ that divides an integer. Such a polynomial is called a $p$-minimal quartic. Let $K$ be a field defined by a $p$-minimal quartic. In this paper, we use $p$-integral bases and introduce the concept of $p$-index forms in order to determine the field index of $K$ via the coefficients of a defining polynomial in terms of certain congruence conditions.

Let $C_n$ denote the cyclic group of order $n$, and let Hol($C_n$) denote the holomorph of $C_n$. In this paper, for any odd integer $m\ge 3$, we find necessary and sufficient conditions on an integer $A$, with $\left|A\right|\ge 3$, such that ${{ℱ}}_{m,A}(x)=x^{2m}+A{x}^m+1$ is irreducible over $\mathbb{Q}$. When $m=q\ge 3$ is prime and ${{ℱ}}_{q,A}(x)$ is irreducible, we show that the Galois group over $\mathbb{Q}$ of ${{ℱ}}_{q,A}(x)$ is isomorphic to either Hol($C_q$) or Hol($C_{2q}$), depending on whether there exists $y\in \mathbb{Z}$ such that $A^2-4=q{y}^2$. Finally, we prove that there exist infinitely many positive integers $A$ such that ${{ℱ}}_{q,A}(x)$ is irreducible over $\mathbb{Q}$ and that $\{1,𝜃,{𝜃}^2,\dots ,{𝜃}^{2q-1}\}$ is a basis for the ring of integers of $K=\mathbb{Q}(𝜃)$, where ${{ℱ}}_{q,A}(𝜃)=0$.

Polynomials of the form $\sum a_{ij}{x}^{p^i+p^j}+L(x)$ are called quadratic polynomials over a finite field ${𝔽}_{p^n}$ and the quadratic polynomials of the form $\sum a_{ij}{x}^{p^i+p^j}$ are called Dembowski–Ostrom polynomials. In this paper, we propose four classes of permutation trinomials of Dembowski–Ostrom type polynomials over finite fields ${𝔽}_{2^n}$ and two classes of quadratic permutation polynomials of the form $\sum a_{ij}{x}^{2^i+2^j}+L(x)$ over ${𝔽}_{2^n}$.

A correspondence is obtained between irreducible cyclic sextic trinomials x⁶ + Ax + B ∈ ℚ[x] and rational points on a genus two curve. This implies that up to scaling, x⁶ + 133x + 209 is the only cyclic sextic trinomial of the given type.