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Is any homomorphism between two isomorphic fields an isomorphism? What I mean is that two fields are called isomorphic if there exist one homomorphism between them . But not sure if existence of one isomorphism means there exist no non-bijective homomorphism between . An explanation or a counter example would help . Please.

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What I mean is that two fields are called isomorphic if there exist one homomorphism between them

You mean if there exists an isomorphism between them. It is not true that a homomorphism between two isomorphic fields needs to be an isomorphism. For example, the natural inclusion $\mathbb{C} \to \overline{ \mathbb{C}(t) }$ is not an isomorphism, but the two fields in question are isomorphic. Perhaps a simpler example is the inclusion $\mathbb{C}(x_2, x_3, x_4, ...) \to \mathbb{C}(x_1, x_2, x_3, ...)$.

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A homomorphism from one field into another is always one-to-one. It need not be onto. But I suspect your "between them" is assumed to imply that it is onto.

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  • $\begingroup$ I don’t understand the downvote here; the answer is terse and lacks a concrete example, but it contains the essential information. $\endgroup$ – Brian M. Scott Jun 2 '13 at 7:21

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