Show that Z7[x]/(x³ + 2) is a field.
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![(2)
Write down a zero divisor in the ring End (Z3 × Z3) (the ring of endomor-
phisms of the abelian group Z3 × Z3).
(3)
Show that Z7[x]/(x³ + 2) is a field.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c7265f2-9dda-4402-bf47-87522a081370%2F6298447b-11f2-4bf0-9b0a-3da85b7af2de%2Ftdrl3hb_processed.jpeg&w=3840&q=75)
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- Find all homomorphic images of the quaternion group.22. Let be a ring with finite number of elements. Show that the characteristic of divides .16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.
- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.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]44. Consider the set of all matrices of the form, where and are real numbers, with the same rules for addition and multiplication as in. a. Show that is a ring that does not have a unity. b. Show that is not a commutative ring.
- Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.A Boolean ring is a ring in which all elements x satisfy x2=x. Prove that every Boolean ring has characteristic 2.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 .
- 11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.