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Galois Groups of Trinomials with Small Coefficients

  • Hilel Garmi
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Abstract

We calculate the Galois group of f = xn + axk + b ∈ ℤ[ x ] over ℚ for all 1 ≤ | ab | ≤ 3, n sufficiently large and k coprime to n. In particular, we prove that the Galois group of f is the maximal one that its factorization type allows. We give a criterion for the Galois group of a factor of a trinomial to be symmetric, with an eye towards future applications. To the best of our knowledge, the only previously known explicit example of polynomials of degree n with bounded coefficients and Galois group Sn is xn − x − 1, while recent results show that the probability that a random polynomial of degree n with reasonably bounded coefficients has Galois group Sn tends to $1$ as n tends to infinity. Using our calculation, one can construct many new such explicit examples.

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Cite

BibTeXRIS
BibTeX
@misc{garmi2026galois,
  title = {{Galois Groups of Trinomials with Small Coefficients}},
  author = {Hilel Garmi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22764},
  eprint = {2610.22764},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22764},
}
RIS
TY  - UNPB
TI  - Galois Groups of Trinomials with Small Coefficients
AU  - Hilel Garmi
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.22764
UR  - https://arxiv.org/abs/2610.22764
AB  - We calculate the Galois group of f = x^n + ax^k + b ∈ ℤ[ x ] over ℚ for all 1 ≤ | ab | ≤ 3, n sufficiently large and k coprime to n. In particular, we prove that the Galois group of f is the maximal one that its factorization type allows. We give a criterion for the Galois group of a factor of a trinomial to be symmetric, with an eye towards future applications. To the best of our knowledge, the only previously known explicit example of polynomials of degree n with bounded coefficients and Galois group S_n is x^n − x − 1, while recent results show that the probability that a random polynomial of degree n with reasonably bounded coefficients has Galois group S_n tends to $1$ as n tends to infinity. Using our calculation, one can construct many new such explicit examples.
ER  -