If K is a field (K,+) it is not isomorphic to (K{0},•).

Let (K,+,•) be a field, let us prove that its additive group (K,+) is not isomorphic to the group of invertibles of K, that is (K{0},•). Let us suppose that there is an isomorphism between them, then, since isomorphisms preserve the order of an element, we would have that:

|{x ∈ K : x+x=0}|=|{x∈ K{0} : x²=1} .

Now if Char(K)=2 we have |{x ∈ K : x+x=0}|≥2 (it has at least 0 and 1 as solutions) while |{x∈ K{0} : x²=1}|=1 (it has 1 and -1 as solutions but -1=1) so Char(K)≠2. But if Char(K)≠2 we have |{x ∈ K : x+x=0}|=1 (0 is the only solution) while |{x∈ K{0} : x²=1}|=2 (has 1 and -1 as solutions) which is absurd. □