Let w € C be a primitive third root of unity. a. Show that Z[w] = {a+bw | a, b Z} is a subring of C. b. Show Z[w] is a Euclidean domain with size function z→ zz.
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![Let w € C be a primitive third root of unity.
a. Show that Z[w] = {a+bw | a, b Z} is a subring of C.
b. Show Z[w] is a Euclidean domain with size function z→ zz.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffaf3b997-d0a9-4686-8fe8-f2a48433a18c%2F4ccfb26f-f5b2-4cfe-871a-25f915a1bef8%2F01p6ta7_processed.jpeg&w=3840&q=75)
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- 9. If denotes the unity element in an integral domain prove that for all .29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .34. If is an ideal of prove that the set is an ideal of . The set is called the annihilator of the ideal . Note the difference between and (of Exercise 24), where is the annihilator of an ideal and is the annihilator of an element of.
- 2. Prove the following statements for arbitrary elements of an ordered integral domain . a. If and then . b. If and then . c. If then . d. If in then for every positive integer . e. If and then . f. If and then .Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]
- Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
- [Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xyLet R be as in Exercise 1, and show that the principal ideal I=(2)={2n+m2|n,m} is a maximal ideal of R. Exercise 1. According to part a of Example 3 in Section 5.1, the set R={m+n2|m,n} is a ring. Assume that the set I={a+b2|aE,bE} is an ideal of R, and show that I is not a maximal ideal of R.
