College Algebra Exam Review 284

College Algebra Exam Review 284 - 294 6.4.12. (a) (b) 6....

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Unformatted text preview: 294 6.4.12. (a) (b) 6. RINGS Show that a 7! Œa=1 is an injective unital ring homomorphism of R into Q.R/. In this sense Q.R/ is a field containing R. If F is a field such that F à R, show that there is an injective unital homomorphism ' W Q.R/ ! F such that '.Œa=1/ D a for a 2 R. 6.4.13. If R is the ring of Gaussian integers, show that Q.R/ is isomorphic to the subfield of C consisting of complex numbers with rational real and imaginary parts. 6.4.14. Let J be an ideal in a ring R. Show that J is prime if, and only if, R=J has no nontrivial zero divisors. (In particular, if R is commutative with identity, then J is prime if, and only if, R=J is an integral domain.) 6.4.15. Every ideal in Z has the form d Z for some d > 0. Show that the ideal d Z is prime if, and only if, d is prime. 6.4.16. Show that a maximal ideal in a commutative ring with identity is prime. 6.5. Euclidean Domains, Principal Ideal Domains, and Unique Factorization We have seen two examples of integral domains with a good theory of factorization, the ring of integers Z and the ring of polynomials KŒx over a field K . In both of these rings R, every nonzero, noninvertible element has an essentially unique factorization into irreducible factors. The common feature of these rings, which was used to establish unique factorization is a Euclidean function d W R n f0g ! N [ f0g with the property that d.fg/ maxfd.f /; d.g/g and for each f; g 2 R n f0g there exist q; r 2 R such that f D qg C r and r D 0 or d.r/ < d.g/. For the integers, the Euclidean function is d.n/ D jnj. For the polynomials, the function is d.f / D deg.f /. Definition 6.5.1. Call an integral domain R a Euclidean domain if it admits a Euclidean function. Let us consider one more example of a Euclidean domain, in order to justify having made the definition. Example 6.5.2. Let ZŒi  be the ring of Gaussian integers, namely, the complex numbers with integer real and imaginary parts. For the ”degree” map d take d.z/ D jz j2 . We have d.zw/ D d.z/d.w/. The ...
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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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