Let K be the set of the following four$-$tuples of elements of GF$(3)$:

$(0,0,0,0),(1,2,1,1),(2,1,2,2),(1,0,0,1),(2,2,1,2),$

$(2,0,0,2),(0,1,2,0),(0,2,1,0),(1,1,2,1).$

Define operations of addition and multiplication on K so that it becomes a field.