The article, 鈥淓isenstein polynomials defining cyclic p-adic fields with minimal wild ramification鈥 appears in a recent issue of the JP Journal of Algebra, Number Theory and Applications.
Associate Professor of Mathematics Chad Awtrey and Haydn Stucker 鈥23 published a research paper in the January 2021 issue of the JP Journal of Algebra, Number Theory and Applications.
The paper, 鈥淓isenstein polynomials defining cyclic p-adic fields with minimal wild ramification鈥, JP Journal of Algebra, Number Theory and Applications, Vol. 49, No. 1, 93-100, (2021), analyzes polynomials with p-adic numbers as coefficients whose symmetries correspond to the rotations of a regular polygon. The collection of symmetries is known as the Galois group of the polynomial, named for French mathematician Evariste Galois (1811-1832). Properties of the polynomial鈥檚 Galois group govern important arithmetic properties of the polynomial鈥檚 roots.
In their research, the authors consider special polynomials known as Eisenstein polynomials, where all coefficients, except the leading coefficient, are multiples of the prime number p. The goal was to determine necessary and sufficient conditions on the coefficients of these polynomials to guarantee their Galois groups are cyclic; that is, they correspond to the rotations of a regular polygon. Previous research by O. Ore and S. Amano studied Eisenstein polynomials of degree p. The authors extended this work to include degrees that are multiples of p but not p^2.