On the height of polynomials that split completely over a fixed number field

Key Points

• The minimal height of polynomials increases exponentially with degree n, confirming a significant result in number theory.
• Exact implied constants were determined for specific fields, including totally real fields and certain quadratic extensions.
• Polynomials of degree n show that roots in specific number fields lead to predictable height growth patterns.
• This research contributes to understanding polynomial behavior in the context of finite extensions over the rationals.