By David M Bressoud; S Wagon
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Extra info for A course in computational number theory
X, are the solutions of X” - s1xn-l + s2xn-2 - s3xn-3 let n + * . + (-l),S, * = 0, 38 The Crearion ojPolynotnial for any integer k;then Around 1666, general formulas for the sum of any power of the solutions were found by Isaac Newton (1642-1727) [45, v. I, p. 5191 (who was probably unaware of Girard’s work: see footnote (12) in [45, v. I, p. 5 181). Newton’s clever observation is that, while the formulas for 01, U Z , . . in terms of s l , . . , s, do not seem to follow a simple pattern, yet there are simple formulas expressing uk in terms of $1, .
I, p. 5191 (who was probably unaware of Girard’s work: see footnote (12) in [45, v. I, p. 5 181). Newton’s clever observation is that, while the formulas for 01, U Z , . . in terms of s l , . . , s, do not seem to follow a simple pattern, yet there are simple formulas expressing uk in terms of $1, . , , sn and 6 1 , . . These formulas can be used to calculate recursively the various d k in ternis of $1, . . , Newton’s formulas are and, generally, These formulas (for k 5 n) were published without proof in “Arithmetica Universalis” (1707) [46, p.
While this tcrminology will he retained in the sequel, it should be observed that, without any other proper definition, to say something is an indeterminate or a variable i s hardly a dcfinition. Moreover, it fosters confusion between the polynomial and the associated polynomial function + + + which maps 2 E A onto P ( x ) = a0 U ~ X . n,z". This same confusion has prompted the use of the term conslant polynomials for the elements of A , considered as polynomials. 16 below, p. 52), it could be harmful when A is finite.