## AN INTRODUCTION TO THE MODERN THEORY OF EQUATIONS |

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Abelian equation Abelian group adjoined adjunction alternating group binomial equations called coefficients complex numbers complex roots cross-ratio cube roots cubic equation cycles cyclic equation cyclic function cyclic group cyclotomic equation discriminant divisible domain 0(a elements equa equal roots equation f(x equation x4 expressed follows fraction Galois domain Galois group Galois resolvent Galois theory given equation group G group of degree Hence identical substitution imprimitive irreducible equation metacyclic multiple roots negative roots normal domain normal equation normal sub-group nth degree nth roots obtain operate permutation polynomial positive roots prime degree prime number primitive roots quadratic quartic quintic rational function rational numbers real roots reciprocal equation reducible replaced roots of f(x roots of unity roots of xn Rule of Signs second term Show that x5 solution Sturm's functions Sturm's Theorem symmetric function symmetric group tion transpositions unaltered vanish variations zero

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Page 24 - Any two sides of a triangle are together greater than the third side.

Page 11 - The coefficient pt of the fourth term with its sign changed is equal to the sum of the products of the roots taken three by three ; and so on, the signs of the coefficients being taken alternately negative and positive, and the number of roots...

Page 9 - VARIATIONS of sign, nor the number of negative roots greater than the number of PERMANENCES. 325. Consequence. When the roots of an equation are all real, the number of positive roots is equal to the number of variations, and the number of negative roots is equal to the number of permanences. For, let m denote the degree of the equation, n the number of variations of the signs, p the number of permanences ; we shall have m=n+p. Moreover, let n' denote the number of positive roots, and p' the number...

Page 51 - ... then the difference between the number of changes of sign in the series when a is substituted for x, and the number when b is substituted for x ejpresses exactly the number of real roots of the equation f (x) = 0 between a and b.

Page 38 - Art. 39 can be applied in general to the solution of this problem, viz. the formation of the equation whose roots are the squares of the differences of every two of the roots of a given...

Page 45 - Q+P is nearly equal to P. By transforming the given equation into another whose roots are the reciprocals of the given equation, the smallest root of the given equation can be found by this process. The idea of considering high powers of other roots negligible in comparison with the like powers of the root which is numerically the largest is due to Newton, who, as we have; seen, determined on this principle an...

Page 44 - If each negative coefficient be taken positively and divided by the sum of all the positive coefficients which precede it, the greatest of all the fractions thus formed increased by unity, is a superior limit of lhe positive roots. Let the equation be f(x) = 0, where f(x) denotes p,l'•K" + plx"~l+p,xn~13-pllx"~a + p,x"~í + ... -px"

Page 87 - PROP. II. — Every rational symmetric function of the roots of an algebraic equation can be expressed rationally in terms of the coefficients. It is sufficient to prove this theorem for integral symmetric functions, since fractional symmetric functions can be reduced to a single fraction whose numerator and denominator are integral symmetric functions. Every integral function of...

Page 51 - ... is equal to the number of real roots between a and b, each multiple root counting only once. EXAMPLES. 1. Find the nature of the roots of the equation x* - 5x> + 9z* - 1x + 2 = 0.

Page 4 - Also, the second term of each partial product may be added to the corresponding term of the dividend, provided we change the sign of the second term of the divisor before multiplying. The work now stands...