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Unformatted text preview: NUMBERS AND SETS EXAMPLES SHEET 4. W. T. G. 1. Is there a field with exactly four elements? Is there one with six elements? 2. Let F be a field. Prove that (- 1)(- 1) = 1 in F . [- 1 is of course defined to be the additive inverse of the multiplicative identity. If you have any difficulty knowing what you are allowed to assume, then change your notation so that it no longer looks familiar.] 3. Let F be an ordered field. Prove that 1 > 0. Prove also that if 0 < x < y , then < 1 /y < 1 /x . 4. Let p be a prime. Prove that the field Z p cannot be made into an ordered field. Prove also that C cannot be made into an ordered field. [That is, in both cases show that there is no relation on the field that satisfies the required axioms.] 5. Prove that there is exactly one way to make Q into an ordered field. [That is, prove that the usual ordering < is the only relation that satisfies the required axioms. To do this, let R be another such relation and prove from the axioms that xRy if and only if...
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- Fall '10