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

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This is a Quantitative Study study.

Thian Tromp

doi:10.48550/arxiv.2509.12064

September 15, 2025Open Access

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

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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.

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Implication

Analysis reveals polynomials' minimal height grows exponentially in degree n, highlighting behavior in specific number fields.

Abstract

Let K/Q be a finite extension. We prove that the minimal height of polynomials of degree n of which all roots are in K^× increases exponentially in n. We determine the implied constant exactly for totally real K and K equal to Q(√-1) or Q(√-3).

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Cite This Study

Thian Tromp (2025) studied this question.

synapsesocial.com/papers/68ed1896f29694dd1da78af5 https://doi.org/10.48550/arxiv.2509.12064

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

Key Points

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Implication

Abstract

Discussion

Cite This Study

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

Key Points

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Implication

Abstract

Discussion

Cite This Study

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