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2019 Essential dimension of inseparable field extensions
Zinovy Reichstein, Abhishek Kumar Shukla
Algebra Number Theory 13(2): 513-530 (2019). DOI: 10.2140/ant.2019.13.513


Let k be a base field, K be a field containing k, and LK be a field extension of degree n. The essential dimension ed(LK) over k is a numerical invariant measuring “the complexity” of LK. Of particular interest is

τ ( n ) = max { ed ( L K ) L K  is a separable extension of degree  n } ,

also known as the essential dimension of the symmetric group Sn. The exact value of τ(n) is known only for n7. In this paper we assume that k is a field of characteristic p>0 and study the essential dimension of inseparable extensions LK. 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 LK, respectively, and τ(n) is replaced by

τ ( n , e ) = max { ed ( L K ) L K  is a field extension of type  ( n , e ) } .

The symmetric group Sn is replaced by a certain group scheme Gn,e over k. This group scheme is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of Sn. Our main result is a simple formula for τ(n,e).


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Zinovy Reichstein. Abhishek Kumar Shukla. "Essential dimension of inseparable field extensions." Algebra Number Theory 13 (2) 513 - 530, 2019.


Received: 28 June 2018; Accepted: 24 December 2018; Published: 2019
First available in Project Euclid: 26 March 2019

zbMATH: 07042068
MathSciNet: MR3927055
Digital Object Identifier: 10.2140/ant.2019.13.513

Primary: 12F05 , 12F15 , 12F20 , 20G10

Keywords: Essential dimension , group scheme in prime characteristic , inseparable field extension

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.13 • No. 2 • 2019
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