The additive and multiplicative groups of a field are never isomorphic

Let F be a field. Prove that as groups, (F,+) and (F^\times, \cdot) are not isomorphic.

First suppose F is finite. Then |F| = n and |F^\times| = n-1 for some natural number n \geq 1. In particular, no bijection \Phi : F \rightarrow F^\times exists, so that these groups cannot be isomorphic.

Suppose now that F is infinite and that \Phi : F \rightarrow F^\times is a field isomorphism. Note that \Phi(0) = 1.

Suppose 1 = -1. Now \Phi(1) = a for some a \in F, and \Phi(-1) = a^{-1}. But also \Phi(-1) = \Phi(1) = a. So a^2 = \Phi(1)\Phi(-1) = \Phi(0) = 1. Thus a^2 = 1, and we have a = \pm 1 = 1. But then \Phi(0) = \Phi(1), and \Phi is not injective, a contradiction.

Suppose instead that 1 \neq -1. Now \Phi(a) = -1 for some a \in F, so that \Phi(2a) = \Phi(a)^2 = 1. Since \Phi is injective, 2a = 0. Thus a = 0, and we have 1 = \Phi(0) = -1, a contradiction.

Thus (F,+) and (F^\times, \cdot) are not isomorphic.

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  • Nick  On January 28, 2011 at 12:44 am

    That should be “of a ring” no?

  • Nick  On January 28, 2011 at 12:45 am

    …nevermind. Egg ON FACE TIMES TEN.

    • nbloomf  On January 28, 2011 at 7:53 am

      No worries. 🙂

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