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Article overview
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Binary quartic forms with bounded invariants and small Galois groups | Cindy
; Tsang
; Stanley Yao Xiao
; | Date: |
23 Feb 2017 | Abstract: | In 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 | Services: | Forum | Review | PDF | Favorites |
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