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Based on the simple and well understood concept of subfields in a finitefield, the technique called `field reduction' has proved to be a very usefuland powerful tool in finite geometry. In this paper we elaborate on thistechnique. Field reduction for projective and polar spaces is formalized andthe links with Desarguesian spreads and linear sets are explained in detail.Recent results and some fundamental ques- tions about linear sets and scatteredspaces are studied. The relevance of field reduction is illustrated bydiscussing applications to blocking sets and semifields.

Field reduction and linear sets in finite geometry