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Unformatted text preview: domain, if 1 2 R . (b) The Gaussian integers are the complex numbers whose real and imaginary parts are integers. Show that the set of Gaussian integers is an integral domain. (c) Show that the ring of symmetric polynomials in n variables is an integral domain. 6.4.8. Show that a quotient of an integral domain need not be an integral domain. 6.4.9. Prove Lemma 6.4.3 . 6.4.10. Show that addition and multiplication on Q.R/ is well dened. This amounts to the following: Suppose a=b a =b and c=d c =d and show that .ad C bc/=bd .a d C c b /=b d and ac=bd a c =b d . 6.4.11. (a) Show that Q.R/ is a ring with identity element 1=1 and D 0=1 . (b) Show that a=b D if, and only if, a D . (c) Show that if a=b , then b=a is the multiplicative inverse of a=b . Thus Q.R/ is a eld....
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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.
 Fall '08
 EVERAGE
 Algebra, Power Series

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