By Harvey Cohn

Eminent mathematician, instructor ways algebraic quantity conception from ancient perspective. Demonstrates how thoughts, definitions, theories have advanced in the course of final 2 centuries. Abounds with numerical examples, over 2 hundred difficulties, many concrete, particular theorems. a number of graphs, tables.

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Thus, iff(Q is a polynomial, with quadratic integers (elementsof E)) as coefficients, 5i s E2 (mod 11)implies f(Ei) =f(12) (mod r). The properties extend to a11rings. Note that d3 E 1 (mod 2) and that %‘? + 1 (mod 2) on the basisof the fact that (a - 1)/2 is an integer but (d3 - 1)/2 is not an integer in each respective field. 1 The definitions of ring and integrul domain are restricted to the context of subsets of the complex numbers. Definitionsof integral domain vary widely in the literature, but we follow the spirit of the original efforts to generalize rational integers.

We generate a11h,h, (different) characters, as we verify below. The reader cari refer to residue classes modulo 8 in Table 2, $1 (above), Here h, = h, = 2, x0, x1, x2, x3 cari easily be identified with the 4 characters xïu, in (4), where u1 = 0, 1 and u2 = 0, 1. Thus, when we have an abelian group of order h (using cyclic structure), we cari show that there are h characters. We cari even see that the character group has the same cyclic structure. ) Specifically, let G = Z(h,) x Z(h,) x . * . x Z(h,) (6) SOthat an arbitrary.

II] 0 + ... + F), s x uoul.. &) = 4toih0P 0 I ui < h,, 0 I ui < hi. * * * e(t,lhJ”~, 0 I ti < h,, using the function (9) e(l) = exp 2rif. , e(E + 1) = e(f), as well as the exponential property e(E + 7) = e(@e(q). Also;e(Q = 1 if and only if t is an integer. It is not obvious that the h characters listed in (8~) are different. For instance, if u. &4 But we need only take a = a, in (7), then the relation (~OU) follows from the obvious result that (with t, = 1, t, = t, = . . t, = 0), (lob) exp 2n-iu,/h, # exp 2n+v,/h,.