On the Discriminant and Index of a Certain Class of Polynomials

Key Points

• The research aims to compute the discriminant of a specific polynomial and determine conditions under which it is monogenic.
• Analyze the polynomial $f(x) = (x^{2}+1)^{n} - ax^{n}$ for irreducibility.
• Compute the discriminant of $f(x)$ and derive conditions for monogenicity based on parameters a and n.
• Describe the primes that divide the index of $f$.
• The discriminant of the polynomial $f(x)$ was computed.
• Necessary and sufficient conditions for $f$ to be monogenic were established based on a and n.
• A complete characterization of the primes dividing the index of $f$ was provided.

Abstract