Unformatted text preview: we see that bd = 1. It is then not diﬃcult to discuss all possibilities. p426, E21 Since x 2 = 1 has at most 2 solutions, 2 is the only element of order 2. That means, any element β which is not 0 , 1 , 2 is of order either 13 or 26. If β has order 13, then β 13 = 1, then (2 β ) 13 = 2 13 β 13 = 2 6 = 1, so 2 β does not have order 13. Therefore it must have order 26. Suppose that β has order 26, then β 13 6 = 1. In addition, ( β 13 ) 2 = 1, so β 13 = 2. Then (2 β ) 13 = 2 13 β 13 = 2 14 = 1. So 2 β has order 13. Note that the converse direct is direct from the previous argument, using the fact that 2 × 2 β = β . 1...
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 Spring '08
 BILLERA
 Math, Algebra, Polynomials, Elementary algebra, Prime number, irreducible quadratic polynomials

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