College Algebra Exam Review 197

College Algebra Exam Review 197 - p ² 1 Proof Let K ±...

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3.6. FINITELY GENERATED ABELIAN GROUPS 207 and forming the direct product of the chosen groups. There are 6 possibil- ities for the elementary divisor decompositions: Z 8 ± Z 3 ± Z 25 ± Z 7 ; Z 8 ± Z 3 ± Z 5 ± Z 5 ± Z 7 ; Z 4 ± Z 2 ± Z 3 ± Z 25 ± Z 7 ; Z 4 ± Z 2 ± Z 3 ± Z 5 ± Z 5 ± Z 7 ; Z 2 ± Z 2 ± Z 2 ± Z 3 ± Z 25 ± Z 7 ; Z 2 ± Z 2 ± Z 2 ± Z 3 ± Z 5 ± Z 5 ± Z 7 : The corresponding invariant factor decompositions are: Z 4200 ; Z 840 ± Z 5 ; Z 2100 ± Z 2 ; Z 420 ± Z 10 ; Z 1050 ± Z 2 ± Z 2 ; Z 210 ± Z 10 ± Z 2 : The group of units in Z N The rest of this section is devoted to working out the structure of the group ˚.N/ of units in Z N . It would be safe to skip this material on first reading and come back to it when it is needed. Recall that ˚.N/ has order '.N/ , where ' is the Euler ' function. The following theorem states that for a prime p , ˚.p/ is cyclic of order p ² 1 . Theorem 3.6.26. Let K be a finite field of order n . Then the multiplicative group of units of K is cyclic of order n ² 1 . In particular, for p a prime number, the multiplicative group ˚.p/ of units of Z p is cyclic of order
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Unformatted text preview: p ² 1 . Proof. Let K ± denote the multiplicative group of nonzero elements of K . Then K ± is abelian of order n ² 1 . Let m denote the period of K ± . On the one hand, m ³ n ² 1 D j K ± j . On the other hand, x m D 1 for all elements of K ± , so the polynomial equation x m ² 1 D has n ² 1 distinct solutions in the field K . But the number of distinct roots of a polynomial in a field is never more than the degree of the polynomial (Corollary 1.8.24 ), so n ² 1 ³ m . Thus the period of K ± equals the order n ² 1 of K ± . But the period and order of a finite abelian group are equal if, and only if, the group is cyclic. This follows from the fundamental theorem of finite abelian groups, Theorem 3.6.2 . n...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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