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29 March 2024
 
  » arxiv » 1702.7407

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Binary quartic forms with bounded invariants and small Galois groups
Cindy ; Tsang ; Stanley Yao Xiao ;
Date 23 Feb 2017
AbstractIn this paper, we enumerate the $operatorname{GL}_2(mathbb{Z})$-equivalence classes of integral binary quartic forms which are fixed under substitution by a particular matrix in $operatorname{GL}_2(mathbb{R})$ which is proportional over $mathbb{R}$ to an integer matrix. In particular, whenever such a form $F$ is irreducible, the Galois group of the splitting field of $F$ is isomorphic to a subgroup of the dihedral group $mathcal{D}_4$ of order eight. We also give a new criterion for when the negative Pell’s equation $x^2 - Dy^2 = -1$ is soluble in integers $x$ and $y$.
Source arXiv, 1702.7407
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