prove that Z*Z is not a vector space over the field R Solution Suppose there is a such a field and proceed by contradiction. I will write multiplication FV, where F is in the field and V is an element of Z. First we rule out the case where the field has characteristic 2. We would have 0 = (1F+1F)1 = 1F1+1F1=2 a contradiction. Now, consider the case where the field does not have characteristic 2. Then there is an element 2F?1in the field, and 1=2F(2?1F1)= 2?1F1 + 2?1F1 Now 2?1F ?Z as it is an element of the vector space, but there is no element a?Z with 2a=1, so we have a contradiction..