Let be a base field, be a field containing , and be a field extension of degree . The essential dimension over is a numerical invariant measuring “the complexity” of . Of particular interest is
also known as the essential dimension of the symmetric group . The exact value of is known only for . In this paper we assume that is a field of characteristic and study the essential dimension of inseparable extensions . Here the degree is replaced by a pair which accounts for the size of the separable and the purely inseparable parts of , respectively, and is replaced by
The symmetric group is replaced by a certain group scheme over . This group scheme is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of . Our main result is a simple formula for .
"Essential dimension of inseparable field extensions." Algebra Number Theory 13 (2) 513 - 530, 2019. https://doi.org/10.2140/ant.2019.13.513