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Publication . Preprint . Article . Other literature type . 2019

Essential dimension of inseparable field extensions

Zinovy Reichstein; Abhishek Kumar Shukla;
Open Access

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

Subjects by Vocabulary

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

Related Organizations
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  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)