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Extensions with Galois group $$2^+S_4*D_8$$ in characteristic 3. (English) Zbl 1153.11056
Let $$2^+S_n$$ be the double cover of $$S_n$$ such that transpositions lift to involutions and products of two disjoint transpositions lift to elements of order $$4$$, and let $$D_8$$ be the dihedral group of order 8. For groups $$G_1$$ and $$G_2$$ with isomorphic centers, denote by $$G_1 * G_2$$ their central product, defined to be the subgroup of $$G_1 \times G_2$$ obtained by identifying the centers of $$G_1$$ and $$G_2$$. Suppose that $$K$$ is a field of characteristic $$3$$. This note describes explicitly all extensions $$L/K$$ with $$\text{Gal}(L/K) \cong 2^+S_4 * D_8$$, thereby improving on previous results of the authors. The calculation is intricate, and relies in part on work of Abhyankar. The case where $$K$$ is the fraction field of the formal power series ring $$k[x_1, x_2, x_3]$$ for some $$k$$ of characteristic $$3$$ is relevant to Abhyankar’s normal crossings local conjecture.
##### MSC:
 11R32 Galois theory
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##### References:
 [1] Abhyankar, S.S., Galois embeddings for linear groups, Trans. amer. math. soc., 352, 3881-3912, (2000) · Zbl 0999.12005 [2] Abhyankar, S.S., Resolution of singularities and modular Galois theory, Bull. amer. math. soc., 38, 131-169, (2001) · Zbl 0999.12003 [3] Crespo, T.; Hajto, Z., On vectorial polynomials and coverings in characteristic 3, Proc. amer. math. soc., 134, 23-29, (2006) · Zbl 1092.12006 [4] Harbater, D.; van der Put, M.; Guralnick, R., Valued fields and covers in characteristic p, Fields inst. commun., 32, 175-204, (2002) · Zbl 1020.14007 [5] Lam, T.Y., The algebraic theory of quadratic forms, (1973), Benjamin-Cummings Publ. Co. Reading, MA · Zbl 0259.10019
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