Subfields of a finite field are completely classified by Theorem 22.1.7. Create a finite field of order 36. Theory says there is a subfield of order 32, since 2∣6. Determine a generator of multiplicative order 8 for the nonzero elements of this subfield, and construct these 8 elements. Add in the field's zero element to this list. It should be clear that this set of 9 elements is closed under multiplication. Absent our theorems about finite fields and cyclic groups, the closure under addition is not a given. Write a single statement that checks if this set is also closed under addition, by considering all possible sums of elements from the set.