## The additive and multiplicative groups of a field are never isomorphic

Let be a field. Prove that as groups, and are not isomorphic.

First suppose is finite. Then and for some natural number . In particular, no bijection exists, so that these groups cannot be isomorphic.

Suppose now that is infinite and that is a field isomorphism. Note that .

Suppose . Now for some , and . But also . So . Thus , and we have . But then , and is not injective, a contradiction.

Suppose instead that . Now for some , so that . Since is injective, . Thus , and we have , a contradiction.

Thus and are not isomorphic.

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## Comments

That should be “of a ring” no?

…nevermind. Egg ON FACE TIMES TEN.

No worries. 🙂