I've begun a solution by finding the orders of the elements and finding the relations between them hoping to come to some contradiction.

However I've come across [this question](https://math.stackexchange.com/questions/42143/can-you-construct-a-field-with-4-elements/), whcih seems to suggest there is a field of order 4? However, I haven't yet covered field extensions. Without field extensions is there no field of order 4?