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"Abstract" fields can be embedded (somehow) into the field of complex numbers. So what are the advantages of considering these abstract fields?

It is said that prime ideals in rings are "behave" like the primes in the ring of integers. A number field contains therefore two objects that are like the prime numbers, namely the prime numbers itsself since it is a field extension of the rational, and the set of prime ideals. This is a little bit strange...

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 What do you mean by an abstract field? Fields with characteristic p certainly cannot be embedded into the field of complex numbers, not to mention fields with strictly larger cardinality than the one of the complex numbers... – ex falso quodlibet Feb 24 2011 at 14:37
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 Kikiriku, please put more thought into your questions before asking on MathOverflow. For more guidance, please read the "how to ask" page, which is linked on the top of this one. – S. Carnahan♦ Feb 24 2011 at 14:51
Not all fields of characteristic 0 can be embedded in the complex field either. – Michael Hardy Feb 25 2011 at 3:58
.....in particular, some fields of characteristic 0 that are countably infinite cannot be embedded in the complex field. – Michael Hardy Feb 25 2011 at 3:59
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 Actually, every countable field of characteristic zero does embed in the field of complex numbers. (Embed it in its algebraic closure and think about transcendence degrees.) – Greg Marks Feb 28 2011 at 21:21

closed as not a real question by Pete L. Clark, S. Carnahan♦ Feb 24 2011 at 14:49

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