Read e-book online Advanced Number Theory PDF

By Harvey Cohn

ISBN-10: 048664023X

ISBN-13: 9780486640235

Eminent mathematician, instructor ways algebraic quantity concept from historic point of view. Demonstrates how recommendations, definitions, theories have developed in the course of final 2 centuries. Abounds with numerical examples, over 2 hundred difficulties, many concrete, particular theorems. a variety of graphs, tables.

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Extra info for Advanced Number Theory

Example text

TO show property (2d), note that since (I[l - 1[‘1)2 2 0, on expanding, we find I[l2 + 1L4y22 2 IEE’I = 2 lN@I. Thus, unless 6 = 0, from the fact that IN(l)1 2 1 it follows that for each 5, IIE/[ 2 1. Furthermore, /I[l/ = II<‘11 and IlaIl = ]a[, with rational a. 1 As a matter of convenience, we define fractional combinations of vectors by saying w1 = p/qw% means qwl = pwz. The fact that this use of fractions is consistent is similar to the fact that the use of fractions in ordinary or modular arithmetic is consistent (sec Chapter 1, $1).

T,], 28 CHARACTERS [Ch. II] where ti are the exponents usedin (5a, b, c) of Chapter 1, $5, with m = p1a1p2a2 ’ * *PS”*. The congruence classesof fi are somewhatvaried but cari be represented symbolically by 0 I ti < h,, (2) wherehi = #p,“s), 2, or +I$(~~I),asthe casemay require (for i > 1, i = 0, i = 1). Of course, 4(m) = h,h, * * . h,, the order of the group of reduced residueclassesM(m), making the usualallowancesfor missingcomponents if8 fm. The additive group of indices may be representedsymbolically in the usual form of (6) in $2 (above) or of (5) in $7, Chapter 1; (3) M(m) = Z(h,J X Z(h,) X .

Hence x1x2 has the properties of a character in (3) and (4). We define xin in like fashion. We cari now have a group of characters X. We cal1 {X~/X~} the (obvious) quotient character using ordinary division: (7) Wx2bi) = xlWx2(aJ In the samespirit we define the unit character by Xo(ai) = 1 for a11a,. (8) In the caseof residue classesunder multiplication, we cari identify a, the group element, with y = a, the residue class,and use x(a) and x(a) interchangeably, with “modula m” and (a, m) = 1 understood.

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Advanced Number Theory by Harvey Cohn

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