A semifield of order 128 and fractional dimension $\frac{7}{3}$ relative to one of its subsemifields

Abstract

In this short note, an example of a semifield of order 128 containing the Galois field ${\mathbb{F}}_8$ is given. Up to our knowledge, this is the first example supporting the following problem by Cordero and Chen (2013): ``There exist semifield planes of order $2^t$, for any integers $t$ relatively prime to 3 that admit semifield subplanes of order $2^3$''.