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# Essential dimension of inseparable field extensions

Let k be a base field, K be a field containing k and L/K be a field extension of degree n. The essential dimension ed(L/K) over k is a numerical invariant measuring "the complexity" of L/K. Of particular interest is $\tau$(n) = max { ed(L/K) | L/K is a separable extension of degree n}, also known as the essential dimension of the symmetric group $S_n$. The exact value of $\tau$(n) is known only for n $\leq$ 7. In this paper we assume that k is a field of characteristic p > 0 and study the essential dimension of inseparable extensions L/K. Here the degree n = [L:K] is replaced by a pair (n, e) which accounts for the size of the separable and the purely inseparable parts of L/K respectively, and \tau(n) is replaced by $\tau$(n, e) = max { ed(L/K) | L/K is a field extension of type (n, e)}. The symmetric group $S_n$ is replaced by a certain group scheme $G_{n,e}$ over k. This group is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of $S_n$. Our main result is a simple formula for \tau(n, e).

Comment: 18 pages

Microsoft Academic Graph classification: Separable space Mathematics Degree (graph theory) Symmetric group Field (mathematics) Combinatorics Separable extension Field extension Group scheme Type (model theory)

Mathematics - Rings and Algebras, Mathematics - Algebraic Geometry, Mathematics - Group Theory, 12F05, 12F15, 12F20, 20G10, Rings and Algebras (math.RA), Algebraic Geometry (math.AG), Group Theory (math.GR), FOS: Mathematics, Algebra and Number Theory, inseparable field extension, essential dimension, group scheme in prime characteristic, 12F05, 12F15, 12F20, 20G10

Mathematics - Rings and Algebras, Mathematics - Algebraic Geometry, Mathematics - Group Theory, 12F05, 12F15, 12F20, 20G10, Rings and Algebras (math.RA), Algebraic Geometry (math.AG), Group Theory (math.GR), FOS: Mathematics, Algebra and Number Theory, inseparable field extension, essential dimension, group scheme in prime characteristic, 12F05, 12F15, 12F20, 20G10

Microsoft Academic Graph classification: Separable space Mathematics Degree (graph theory) Symmetric group Field (mathematics) Combinatorics Separable extension Field extension Group scheme Type (model theory)

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MR 1009787 [Kar89] Gregory Karpilovsky, Topics in field theory, North-Holland Mathematics Studies, vol. 155, NorthHolland Publishing Co., Amsterdam, 1989, Notas de Matema´tica [Mathematical Notes], 124.

MR 982265 [Knu91] Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 294, Springer-Verlag, Berlin, 1991, With a foreword by I. Bertuccioni. MR 1096299 [Lan02] Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556 [Mer09] Alexander S. Merkurjev, Essential dimension, Quadratic forms-algebra, arithmetic, and geometry, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009, pp. 299-325. [OpenAIRE]

MR 2537108 [Mil17] J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017, The theory of group schemes of finite type over a field.

MR 3729270 [MR09] Aurel Meyer and Zinovy Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra Number Theory 3 (2009), no. 4, 467-487.

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MR 0284421 [Rei10] Zinovy Reichstein, Essential dimension, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 162-188. MR 2827790 [RV06] Zinovy Reichstein and Angelo Vistoli, Birational isomorphisms between twisted group actions, J.

Lie Theory 16 (2006), no. 4, 791-802. MR 2270660 [RV18] , Essential dimension of finite groups in prime characteristic, C. R. Math. Acad. Sci.

Paris 356 (2018), no. 5, 463-467. MR 3790415 [RY00] Zinovy Reichstein and Boris Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, Canad. J. Math. 52 (2000), no. 5, 1018-1056, With an appendix by J´anos Koll´ar and Endre Szabo´. MR 1782331 [SdS00] Pedro J. Sancho de Salas, Automorphism scheme of a finite field extension, Trans. Amer. Math.

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Let k be a base field, K be a field containing k and L/K be a field extension of degree n. The essential dimension ed(L/K) over k is a numerical invariant measuring "the complexity" of L/K. Of particular interest is $\tau$(n) = max { ed(L/K) | L/K is a separable extension of degree n}, also known as the essential dimension of the symmetric group $S_n$. The exact value of $\tau$(n) is known only for n $\leq$ 7. In this paper we assume that k is a field of characteristic p > 0 and study the essential dimension of inseparable extensions L/K. Here the degree n = [L:K] is replaced by a pair (n, e) which accounts for the size of the separable and the purely inseparable parts of L/K respectively, and \tau(n) is replaced by $\tau$(n, e) = max { ed(L/K) | L/K is a field extension of type (n, e)}. The symmetric group $S_n$ is replaced by a certain group scheme $G_{n,e}$ over k. This group is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of $S_n$. Our main result is a simple formula for \tau(n, e).

Comment: 18 pages