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Revisiting Kneser’s theorem for field extensions. (English) Zbl 1413.11115
For non-empty finite subsets \(S\) and \(T\) of an abelian group \(G\), define the sumset \(S+T\) by \(\{s+t\; :\; s\in S, \; t\in T\}\). Then, by the classical result of M. Kneser [Math. Z 66, 88–110 (1956; Zbl 0073.01702)], \(| S+T| \geq| S| +| T| -1\) or there exists a subgroup \(H\neq \{0\}\) of \(G\) such that \(S+T+H=S+T\). X.-D. Hou, K. H. Leung, and Q. Xiang [J. Number Theory 97, 1–9 (2002; Zbl 1034.11020)] generalized Kneser’s theorem to field extensions in the following form. Let \(F\) be a field, \(L/F\) a field extension, and let \(S\) and \(T\) be finite dimensional subspaces of \(L\) over \(F\). Denote by \(ST\) the \(F\)-linear span of the set of products \(st, \; s\in S, \; t\in T\). Suppose that every algebraic element in \(L\) is separable over \(F\). Then \(\dim ST \geq \dim S + \dim T - 1\) or there exists a subfield \(K\), \(F \subsetneq K \subset L\), such that \(STK=ST\). X.-D. Hou [Linear Algebra Appl. 426, 214–227 (2007; Zbl 1132.12003)] conjectured that the above assertion holds without the separability assumption.
In the paper under review the authors prove Hou’s conjecture by giving an alternative proof without the separability assumption, moreover, with \(K\) depending only on one of the factors. The result is a transposition to the extension field setting of a theorem of É. Balandraud [Ann. Inst. Fourier 58, 915–943 (2008; Zbl 1143.11039)].

MSC:
11P70 Inverse problems of additive number theory, including sumsets
11T99 Finite fields and commutative rings (number-theoretic aspects)
12F99 Field extensions
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