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November 27th, 2007, 12:46 PM   #1
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rings and fields

Let R be a ring, F a subring that is a field. Let w be an element in R that is not in F. Prove that all the elements a + bw with a and b in F are distinct.
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December 6th, 2007, 02:49 AM   #2
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Re: rings and fields

Quote:
Originally Posted by just17b
Let R be a ring, F a subring that is a field. Let w be an element in R that is not in F. Prove that all the elements a + bw with a and b in F are distinct.
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If a + bw = a' + b'w, with a, a', b, b', then w= (a'-a)/(b-b'), and this belongs to F ==> it HAS to be that b = b', and then a = a'.

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Tonio
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