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On the Height of Generators of Galois Extensions with Galois Group An

Q: Can the techniques used in this paper be extended to study the height of generators for Galois extensions with other Galois groups beyond An and Sn?

This is a very interesting question that the paper hints at towards the end. While the results focus on $A_n$ and $S_n$, some techniques might be adaptable to other Galois groups, but it's not straightforward and depends on the group's structure. Here's why: Exploiting Symmetry: The core idea in the paper relies heavily on the high degree of symmetry in $A_n$ and $S_n$. Lemmas like 2.1 and 3.5, and the use of convexity in Proposition 3.6, all hinge upon how these groups act on the set of conjugates. For other groups, similar arguments might work if one finds actions that allow for useful averaging or bounding. Transitivity is not Enough: Transitivity (Lemma 2.1) is used extensively, but it's a common property of Galois groups acting on conjugates. The challenge lies in going beyond transitivity to find group actions that provide finer control over the expressions for the height. Specific to Alternating/Symmetric: Results like Proposition 3.1, crucial for proving the 'if and only if' condition for generators, are quite specific to how roots of unity behave under the action of cycles and transpositions present in $A_n$ and $S_n$. Finding analogous results for other groups would be key. Potential Directions: Groups with Large Actions: Groups embedding naturally into $S_n$ with "large" orbits when acting on subsets of {1,...,n} might be approachable. Solvable Groups: The paper mentions the dihedral group case being handled in other work. Generalizing to other solvable groups, where one could potentially exploit towers of extensions, is a possibility. In summary, extending these techniques requires carefully analyzing the target Galois group's structure to identify exploitable properties beyond mere transitivity. It's likely a case-by-case exploration.

Q: What if we relax the condition that β is a unit in Theorem 1.4? Does the height of the generator still tend to infinity as n increases?

Relaxing the unit condition in Theorem 1.4 makes the problem significantly more challenging. The proof heavily relies on β being a unit to establish the lower bound on the height. Here's why: Dobrowolski's Theorem: The final step in the proof uses Dobrowolski's theorem, which provides a lower bound for the Mahler measure (and hence the height) of algebraic numbers that are not roots of unity. If β is not a unit, we can't directly apply this theorem. Controlling the Norm: When β is a unit, its norm is 1, which greatly simplifies the expressions involving the Mahler measures of α and β. Without this constraint, the norm of β could potentially counteract the growth coming from the exponents in the expression for α, making it difficult to prove that the height tends to infinity. Potential Approaches: Stronger Lower Bounds: One would need to find alternative lower bounds for the Mahler measure of algebraic numbers that are not roots of unity but don't require the unit condition. Such bounds are generally harder to come by. Analyzing the Norm: A more delicate analysis would be required to understand how the norm of β interacts with the exponents in α and whether one can still salvage a result showing the height grows. In conclusion, relaxing the unit condition is non-trivial. It would require either finding new techniques for bounding the Mahler measure or a much more intricate analysis of the interplay between the norm of β and the structure of the generator α.

Core Concepts

This article investigates the height of generators of Galois extensions of the rational numbers, specifically focusing on extensions with the alternating group An as their Galois group, and proves that certain classical constructions of these generators result in their height tending to infinity as n increases.

Abstract

Bibliographic Information: Jenrin, J. (2024). On the height of some generators of Galois extensions with big Galois group [Preprint]. arXiv:2403.00500v2
Research Objective: This paper aims to explore the behavior of the height of generators of Galois extensions of the rational numbers, particularly those with Galois groups equal to the alternating group An, and to determine if the height of these generators tends to infinity as n increases.
Methodology: The author employs tools from Galois theory, including the properties of the alternating group An, and concepts from number theory, such as the Mahler measure and the logarithmic Weil height of algebraic numbers. The proofs rely on analyzing the action of the Galois group on the generators and establishing lower bounds for their heights.
Key Findings:
- The paper proves that for generators of Galois extensions with Galois group An, constructed as products or linear combinations of conjugates of an algebraic number, their height tends to infinity as n increases.
- This result holds under specific conditions on the algebraic number and the coefficients used in the construction of the generators.
- The paper provides explicit lower bounds for the height of these generators, demonstrating the growth rate as a function of n.
Main Conclusions: The findings provide evidence supporting an extension of a conjecture by Amoroso, originally formulated for Galois extensions with the symmetric group Sn, to extensions with Galois groups containing An. The results suggest that the height of generators of such extensions is significantly influenced by the structure of the Galois group.
Significance: This research contributes to the field of number theory, specifically to the study of Lehmer's conjecture, which posits a lower bound for the height of algebraic numbers. The paper's focus on Galois extensions with specific Galois groups provides valuable insights into the properties of these extensions and the behavior of their generators.
Limitations and Future Research: The results are proven for specific constructions of generators of Galois extensions with Galois group An. Further research could explore whether similar results hold for other types of generators or for Galois extensions with different Galois groups. Additionally, investigating the optimal lower bounds for the height of these generators remains an open question.

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Stats

For any ϵ > 0, there exists a positive effective constant c(ǫ) such that, for every α ∈SGal of degree d over Q and not a root of unity, one has h(α) ≥c(ǫ)d−ǫ.
If α is a generator of Q(β1, . . . , βn)/Q and β is a unit, then h(α) ≥(1 + g(n))√2n200π(log(log(n))log(n))3 where g(n) tends to 0 as n tends to inﬁnity.
If α is a generator of Q(β1, . . . , βn)/Q, then h(α) ≥ 1/240 log(n/9).

Quotes

"A set of algebraic numbers which has attracted the attention in this respect in recent years, is the following SGal = {α ∈Q | Q(α)/Q is Galois }."
"A result this strong might prompt the question, of whether the set SGal satisfies Property (B) or not."
"The goal of this article is to investigate whether some analogue of Theorem 1.1 holds for other groups."
"We notice that our results potentially support an extension of Conjecture 1.2 to generators of Galois extensions with Galois group containing An."

Key Insights Distilled From

On the height of some generators of galois extensions with big galois group

by Jonathan Jen... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2403.00500.pdf

On the height of some generators of galois extensions with big galois group

Deeper Inquiries

Can the techniques used in this paper be extended to study the height of generators for Galois extensions with other Galois groups beyond An and Sn?

This is a very interesting question that the paper hints at towards the end. While the results focus on  $A_n$ and $S_n$, some techniques might be adaptable to other Galois groups, but it's not straightforward and depends on the group's structure. Here's why:

Exploiting Symmetry: The core idea in the paper relies heavily on the high degree of symmetry in $A_n$ and $S_n$.  Lemmas like 2.1 and 3.5, and the use of convexity in Proposition 3.6, all hinge upon how these groups act on the set of conjugates. For other groups, similar arguments might work if one finds actions that allow for useful averaging or bounding.

Transitivity is not Enough:  Transitivity (Lemma 2.1) is used extensively, but it's a common property of Galois groups acting on conjugates. The challenge lies in going beyond transitivity to find group actions that provide finer control over the expressions for the height.

Specific to Alternating/Symmetric:  Results like Proposition 3.1, crucial for proving the 'if and only if' condition for generators, are quite specific to how roots of unity behave under the action of cycles and transpositions present in $A_n$ and $S_n$.  Finding analogous results for other groups would be key.
Potential Directions:

Groups with Large Actions: Groups embedding naturally into $S_n$ with "large" orbits when acting on subsets of {1,...,n} might be approachable.
Solvable Groups:  The paper mentions the dihedral group case being handled in other work.  Generalizing to other solvable groups, where one could potentially exploit towers of extensions, is a possibility.
In summary, extending these techniques requires carefully analyzing the target Galois group's structure to identify exploitable properties beyond mere transitivity. It's likely a case-by-case exploration.

What if we relax the condition that β is a unit in Theorem 1.4? Does the height of the generator still tend to infinity as n increases?

Relaxing the unit condition in Theorem 1.4 makes the problem significantly more challenging. The proof heavily relies on β being a unit to establish the lower bound on the height. Here's why:

Dobrowolski's Theorem: The final step in the proof uses Dobrowolski's theorem, which provides a lower bound for the Mahler measure (and hence the height) of algebraic numbers that are not roots of unity. If β is not a unit, we can't directly apply this theorem.

Controlling the Norm: When β is a unit, its norm is 1, which greatly simplifies the expressions involving the Mahler measures of α and β. Without this constraint, the norm of β could potentially counteract the growth coming from the exponents in the expression for α, making it difficult to prove that the height tends to infinity.
Potential Approaches:

Stronger Lower Bounds: One would need to find alternative lower bounds for the Mahler measure of algebraic numbers that are not roots of unity but don't require the unit condition. Such bounds are generally harder to come by.
Analyzing the Norm:  A more delicate analysis would be required to understand how the norm of β interacts with the exponents in α and whether one can still salvage a result showing the height grows.
In conclusion, relaxing the unit condition is non-trivial. It would require either finding new techniques for bounding the Mahler measure or a much more intricate analysis of the interplay between the norm of β and the structure of the generator α.

How does the concept of height relate to other areas of mathematics, such as algebraic geometry or cryptography, and what are the potential implications of these connections?

The concept of height, while originating in number theory, has deep connections and implications for various areas of mathematics:
Algebraic Geometry:

Arithmetic Geometry: Height functions are fundamental in Diophantine geometry, which studies integer or rational solutions to polynomial equations. They provide a measure of the "arithmetic complexity" of points on algebraic varieties.
Finiteness Theorems:  A key application is in proving finiteness results, like the Mordell-Weil theorem, which states that the group of rational points on an elliptic curve over a number field is finitely generated. Heights are used to show that sets of points with bounded height are finite.
Arakelov Theory: This theory generalizes classical intersection theory to work over arithmetic surfaces. Heights play a crucial role in defining intersection numbers in this setting.
Cryptography:

Elliptic Curve Cryptography (ECC): The security of ECC relies on the difficulty of the Discrete Logarithm Problem (DLP) on elliptic curves. The height of points on the curve is related to the complexity of computing the DLP. Points with smaller height are generally easier to attack, so understanding height distribution is important for choosing secure curves.
Pairing-Based Cryptography:  Bilinear pairings on elliptic curves are used in various cryptographic protocols. The efficiency of these pairings depends on the height of certain points on the curve, so optimizing heights is crucial for practical implementations.
Other Areas:

Dynamical Systems: Heights appear in the study of arithmetic dynamics, where one iterates rational functions over number fields. They help analyze the growth and distribution of orbits under iteration.
Transcendental Number Theory: Heights are used to measure the "complexity" of transcendental numbers.  Results like the Lehmer conjecture, mentioned in the paper, have implications for understanding the distribution of algebraic numbers within the complex numbers.
Implications of Connections:

Cross-Fertilization: The interplay between height and these areas leads to a rich exchange of ideas and techniques. For example, tools from algebraic geometry can be used to prove results about heights, which in turn have applications in cryptography.
New Directions:  These connections often open up new research directions. For instance, the study of heights in the context of higher-dimensional varieties or over function fields is an active area of research.
Practical Applications:  Understanding heights has practical implications for cryptography, as mentioned above, in terms of choosing secure parameters and optimizing implementations.
In conclusion, the concept of height bridges various mathematical disciplines, leading to a deeper understanding of number theory, algebraic geometry, cryptography, and beyond. These connections continue to inspire new research and have practical implications for real-world applications.