Advanced Number Theory by Harvey Cohn

By Harvey Cohn

Eminent mathematician, instructor techniques algebraic quantity idea from old viewpoint. Demonstrates how suggestions, definitions, theories have developed in the course of final 2 centuries. Abounds with numerical examples, over two hundred difficulties, many concrete, particular theorems. a number of graphs, tables.

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X(a,), (4 with the properties Aa,) # 0, (3) (4) x(aJxW = x(w). It is easily seenthat x(e) = 1, using a,e = a, in (4). Furthermore, if h is the order of the group, then ah = e, for any elementa of the group. en 22 ab instead of a ~3 b. [St%. 11 23 DEFINITIONS for an appropriate t. , h - 1. Thus the number of possible characters x determined by the values (2) is at most hh. Actually, there are precisely h characters, as we shall soon see. ) Theproduct of two different characters, denoted by {x1x2}, is a character if we deiîne for an arbitrary group eiement a, (6) bhx21@) = xl(4x2W using ordinary multiplication.

Of course, (10) is Dirichlet’s lemma for m in the convenient form (2). EXERCISE 11. Showthat the only m for which aIl charactersare real are 1, 2 (trivial), and 3, 4, 6, 8, 12,24. In fact, this is equivalentto the statementthat aII hi = 2 in (2) of 03 (above). ) EXERCISE 12. Justify the “if and only if” statementsin the proof of (66) and (6~). “) EXERCISE 13. Justify the factor 2(4-z”1’in (10) and Write out the similar equationif m is odd or if 2 11 m and 2211 m. 1SeeExercise12 below. chapter III Some algebraic concepts 1.

3) 3. Modules We define a module as a set of quantities closed under addition and subtraction. , to denote modules. The various vector sets used earlier [such as ind y in Chapter 1 and (E, E’) of $2 above] clearly satisfy the definition of module. For the most important applications we generalize (2b) of the preceding section : We consider combinations of a finite set of vectors’ Vi, (1) u = xlvl + xzv, + * * - + xsvs, where the xi range over a11integers. The set of these II forms a module 9JI and the vectors Vi, V,, * * .

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