F q -pseudoreguli of PG ( 3 , q 3 ) and scattered semifields of order q 6

In this paper, we study rank two semifields of order q that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in [12] for q ≡ 1(mod 3). Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with q ≡ 1(mod 3); for any q = 2, such that h ≡ 1(mod 3) and for any q = 3 with h ≡/ 0 (mod 3). Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields.