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 deﬁned. This amounts to the following: Suppose a=b ³ a =b and c=d ³ c =d and show that .ad C bc/=bd ³ .a d C c b /=b d and ac=bd ³ a c =b d . 6.4.11. (a) Show that Q.R/ is a ring with identity element Œ1=1Ł and D Œ0=1Ł . (b) Show that Œa=bŁ D if, and only if, a D . (c) Show that if Œa=bŁ ¤ , then Œb=aŁ is the multiplicative inverse of Œa=bŁ . Thus Q.R/ is a ﬁeld....
View
Full Document
 Fall '08
 EVERAGE
 Algebra, Power Series, Complex number, Integral domain, Ring theory, RŒŒx

Click to edit the document details