Symplectic groups as Galois groups
Let \(\mathbb{F}\) be a field containing the field of order \(q\), let \(x\) and \(t\) be indeterminates, let \(m\) be an integer greater than 1, and let \[ \widehat{f}(Y)= Y^{q^{2m}}+ t^q Y^{q^{m+1}}+ xY^{q^m}+ tY^{q^{m-1}}+Y. \] In [\textit{S. S. Abhyankar}, Proc. Am. Math. Soc. 124, 2977--2991 (1996; Zbl 0866.12005)], it was shown that, for \(m>2\), the Galois group of \(\widehat{f}(Y)\) over \(\mathbb{F}(x,t)\) is the symplectic group \(\text{Sp}(2m,q)\) in its natural action on vectors, and in [\textit{S. S. Abhyankar} and \textit{P. A. Loomis}, Contemp. Math. 245, 63--76 (1999; Zbl 0982.12002)] this was extended to \(m=2\). In this paper we show that much the same is true if we set \(t=1\); if \[ f(Y)= Y^{q^{2m}}+ Y^{q^{m+1}}+ xY^{q^m}+ Y^{q^{m-1}}+ Y \] and \(m>3\), then the Galois group of \(f(Y)\) over \(\mathbb{F}(x)\) is the symplectic group \(\text{Sp}(2m,q)\) in its natural action on vectors.
scientific article; zbMATH DE number 1471029
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