How to show that $(16, 45)$ is a Dickson's pair pair?
Where the semi-fields of order $q ^ n$ with center $GF(q)$ constructed by the Dickson – Zassenhaus construction are called Dickson semi-fields, the indicated pair of numbers $(q,n)$ is called the Dickson pair. The class of all Dickson semi-fields of order q n with center $GF (q), q = p ^ l$, is denoted by $DF(q, n)$.
How to construct a grid of a under-near-field of an near-Dickson field Q ∈ DF (16, 45)?
Let $q = p ^ l$, where p is a prime, and let n be such an integer the number that all its prime divisors divide the number $q - 1$ and $n -≡ 0 \text{ (mod } 4)$ if $q ≡ 3 \text{ (mod } 4)$. Then for $r = ln$ we can construct an almost-field K consisting of $p ^ r$ as follows elements based on the Galois field $GF(pr)$.
Does anyone have any suggestions on how to prove that semi-field of order $8$ is field?