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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**Example text**

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,, * * .