Essential dimension of inseparable field extensions
| Autor: | Reichstein, Zinovy, Shukla, Abhishek Kumar |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Alg. Number Th. 13 (2019) 513-530 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/ant.2019.13.513 |
| Popis: | 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 |
| Databáze: | arXiv |
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