| 1. | Such real closed fields that are not Archimedean, are non-Archimedean ordered fields.
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| 2. | With this identification, the ordered field "'* R "'of hyperreals is constructed.
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| 3. | This is because any square in an ordered field is at least, but.
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| 4. | Like all ordered fields that properly include "'R "', this field is non-Archimedean.
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| 5. | In fact, it is a very special ordered field : the biggest one.
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| 6. | For example, it is not enough to construct an ordered field with infinitesimals.
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| 7. | An ordered field necessarily has characteristic 0 since the elements necessarily are all distinct.
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| 8. | The notion arose from the theory of ordered groups, ordered fields, and local fields.
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| 9. | Conversely, it is known that any weakly o-minimal ordered field must be real closed.
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| 10. | Dubois showed in 1967 that the answer is negative in general for ordered fields.
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