**Problem.** Prove that the additive group and the multiplicative group of a field are never isomorphic.

**Solution. **Let be a field. Let and be the additive and the multiplicative group of respectively. Suppose that Then the number of solutions of the equation in must be equal to the number of solutions of in Now, if then has only one solution in but will have solutions in If then has exactly two solutions in but has only one solution in Contradiction!

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if the additive and multiplicative groups of a field are isomorphic then the field is a reductive Lie algebra, because something false implies anything

Well, I’m not sure what you mean by that. Any associative commutative algebra (over a field) is a reductive Lie algebra because by defining the algebra becomes an abelian, and hence reductive, Lie algebra.

Hi–

How do you prove that if the additive and multiplicative groups of a field are isomorphic then is infinite.

Hi

Well, the reason is clear: two isomorphic groups must have the same cardinality. If F is finite, then the additive group of F will have |F| elements but the multiplicative group of F will have

|F| – 1 elements and so they can’t be isomorphic. It’s worth mentioning that my proof works perfectly even without this fact.

(I just removed that part and you should take a look at the proof again!)