On spectra and minimal polynomials in finite semifields

We apply the notion of a one-side-ordered minimal polynomial to investigations in finite semifields. A proper finite semifield has non-associative multiplication, that leads to the anomalous properties of its left and right spectra. We obtain the sufficient condition when the right (left) order of a semifield element is a divisor of the multiplicative loop order. The interrelation between the minimal polynomial of non-zero element and its right (left) order is described using the spread set. This relationship fully explains the most interesting and anomalous examples of small-order semifields.