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Proceedings of the Canadian Institute (Volume 2)

Proceedings of the Canadian Institute (Volume 2)


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1884. Excerpt: ... Therefore, i is a root of the rational equation p (x) = 0, being at the same time a root oi the rational (see 11) equation xp (x) = 0, whose roots are the primitive iVth roots of uuity. Hence ty (x) and p (x) have a common measure. But by 11, ip (x) is irreducible. Therefore it is a measure of p (x); and the roots of the equation f/ tx) = 0 are roots of the equation p (x) = 0. Therefore, /(w,0, etc.) = ar9(t) = 0. 13. Another corollary is, that if /(wi 0, etc.) = hw + Ji2(oi +.. + hn = 0, where h, As, etc., are clear of cux, the coefficients hi, hi, etc., are all equal to one another. For, by 12, because f (an, 0, etc.) = 0, /(m, 0i, etc.) = 0. Therefore w / (to, 0U etc.) = 0. In to i f (to, 0i, etc.) give to successively its n--1 different values. Then, in addition, nAj--A, + A, +.. + An. Similarly, n = A, + A, +.. + A . . A, = hr In like manner all the terms hi, hi, etc., are equal to one another. 14. Proposition II. If the simplified expression rj, one of the particular cognate forms of B, be a root of the rational equation F (x) = 0, all the particular cognate forms of R are roots of that equation. For, let ri be a particular cognate form of R. By 12, the law to be established holds when there are no surds in r that are not roots of unity. It will be kept in view that, according to 1, when roots l of unity are spoken of, such roots are meant as 1 m, m being a prime number. Assume the law to have been found good for all expressions that do not involve more than n--1 distinct surds that are net roots of unity; then, making the hypothesis that ri involves not more than n distinct surds that are not roots of unity, the law can be shown l still to hold; in which case it must hold universally. For, let J not...


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Product Details
  • ISBN-13: 9781235664687
  • Publisher: General Books
  • Publisher Imprint: General Books
  • Height: 246 mm
  • No of Pages: 162
  • Spine Width: 9 mm
  • Width: 189 mm
  • ISBN-10: 1235664686
  • Publisher Date: 01 Feb 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 299 gr

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