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Full text of "History Of The Theory Of Numbers - I"

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246 HISTORY or THE THEORY OF NUMBERS. [CHAP, vm the differences of the roots of/(x2)=0 is a quadratic non-residue or residue of p [Pellet81]. Let A! be the like product for f(x). Then A = (-l)"221' /(COA]2. Hence /(az2-f-6) is irreducible if ( !)'/(&)/a* is a quadratic non-residue and then/(ao^+&) is irreducible modulo p for every i and even v. O. H, Mitchell84 gave analogues of Fermat's and Wilson's theorems modulis p (a prime) and a function of x. A. E. Pellet85 considered the exponent n to which belongs the product P of the roots of a congruence F(x)z=Q of degree v irreducible modulo p. If q is a prime factor of n, F(xg) is irreducible or the product of q irreducible factors of degree v modulo p according as q is not or is a divisor of (p l)/n. In particular, F(xx) is irreducible modulo p if, for v even, X contains only prime factors of n not dividing (p l)/n; for v odd, we can use the factor 2 in X only once if p = 4ra+l. Let i be a root of F(x)z=Q, i± a root of an irreducible congruence Jp1(x)=0 (mod p) of degree j^ prime to v. Then t'z*! is a root of an irreducible congruence 0(x) = 0 (mod p) of degree w^. F(x) belongs to the exponent Nn modulo p, where n is prime to (pv 1) -7- -j(p l)^f. Let £1 be a prime factor of N not dividing p 1. Then <?(z*0 is irreducible or decomposes into q^ irreducible factors of degree vvl according as qi is not or is a divisor of (p"I)/N. Thus G(zx) is irreducible if X contains only prime factors of N dividing neither p 1 nor (p"~l)/N. 0. H. Mitchell86 defined the prime totient of f(x) to mean the number of polynomials hi x, incongruent modulo p, of degree less than the degree of (x) and having no factor hi common with / modulo p. Those which contain S, but no prune factor of / not contained hi S, are called $-totitives of/. C. Dina87 proved known results on congruences modulis p and F(x). A. E. Pellet88 proved that, if fj, distinct values are obtained from a rational function of x with integral coefficients by replacing x successively by the m roots of an irreducible congruence modulo p, then JJL is a divisor of m and these M values are the roots of an irreducible congruence. Thus if A is a rational function of any number of roots of congruences irreducible modulo p, and v is the number of distinct values among A, Ap, Ap\..., these values satisfy an irreducible congruence modulo p. If A belongs to the exponent n modulo p, then v is the least positive integer for which p"= 1 (mod n). He proved a result of Serret's74 stated in the following form: If, in an irreducible function F(x) modulo p of degree v and exponent n, x is replaced by rex, where X contains only primes dividing n, then F(x*) is a product of irreducible factors of degree vq and exponent nX, where q is the least integer for which p"9=l (mod nX). He proved the first theorem of Pellet85 and the last one of Pellet.77 "Johns Hopkins University Circulars, 1, 1880-1, 132. "Comptes Rendus Paris, 93, 1881, 1065-6. Cf. Pellet.88 "Amer. Jour, Math., 4, 1881, 25-38. »7Giornale di Mat., 21, 1883, 234-263. For comments on 263-9, see the chapter on quadratic