Let Z,[i] = {a + bi |a, bƐZ,} (see Example 9 in Chapter 13). Show that the field Z,[i] is ring-isomorphic to the field Z,[r]/(x? + 1).
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![Let Z,[i] = {a + bi |a, bƐZ,} (see Example 9 in Chapter 13). Show
that the field Z,[i] is ring-isomorphic to the field Z,[r]/(x? + 1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fd06d49-08c1-4c69-9a5c-c5b264d85373%2F3b9b3aed-e8e8-45e2-a971-47d3e3e54b34%2Fu5txrs.jpeg&w=3840&q=75)
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- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]
- Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
- Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.