à) In Z18, find all proper ideals: >) Find all the elements (cosets) of Zg/I, where I is the ideal generated by {[0], [4]}.
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![4. a) In Z18, find all proper ideals:
4. b) Find all the elements (cosets) of Zg/I, where I is the ideal
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- 1) Find gcd (272, 1479) and find the integers x, y such that gcd(272, 1479) 1479y = 272x +(B) Find all maximal ideals of (Z8, +8,8).Let a, b e Z and let d := (a, b). Prove that { ia + jb : i, j e Z}= {kd : k E Z}. In other words, prove that the integers of the form ia + jb are precisely the multiples of (a, b). (This result will be very important when we study the ideals of Z.) 2.
- 6. Find all the roots of the following polynomials: a) p(r) = a* – 3.r +x² + 7x – 30 and b) p (r) = r* - 4.r + 3² + 14:r + 26. Hint. There is a mathematical statement that says if p (r) is a polynomial with real coeffi- cients and a + bi is a complex root of p (x), then a – bi is also a root of p (r). If, respectively, 1- 2i and 3+ 2i are roots of the polynomials a) and b) given above, then applying the statement will give a second root. Find the other roots. Do not use any caleulating devices to find the other roots. Show your work for finding the other roots. 1 Bonus: Let A = {-3, –2, –1,0, 1,2,3} and B = {-10, –9, ...,9, 10} be sets. A function f: A - B is defined by f (n) = { n2, if ne{-1,0,1) -n, if ne{-2,2} n2-9, if ne(-3,3) a) Write the function f (n) as a set of ordered pairs. b) What is the domain, codomain, and range of the function? c) What is f (X) where X = {-3, –2, –1}?| %).· lI. A: Y (b) (c) (d) 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d) just ideal (b) (c) (d) 3. In the ring (Z12, +12,·12 ), the following ideal is maximal (a) (2) (b) (3) (c) (a) and (b) (d) (a) or (b)Q2 (b): let A and B two ideals with unity of Commutative AtB = IR sing IR Such that Prove A.B = ANB.
- THE RADICAL OF AN IDEAL IS ALSO AN IDEAL. PROVE THIS THEOREM CLEARLY AND TYPE ANSWER.Let f=z" + an-12¹-1 + ... + a₁ € C[z] be a polynomial of degree n ≥ 1. Which of the factorisations below must exist? Select one or more: a. f=(z-C₁)(z-C₂)(z- Cn), where C₁, ..., Cn are complex numbers b. f= (z-C₁)(z-C₂)(z-Cn), where C₁, . ...., Cn are different complex numbers c. f=g₁(z) gk(z), where g₁, . ...., 9k € C[z] each have exactly one root, and all these roots are different d. f = (z² + r₁Z+S₁)... (z² + rkz + Sk), where r₁, ..., rk and S₁, ..., Sk are real numbers e. f=g₁(z) gk(z), where g₁, , ..., 9k € C[z] each have exactly one root, and their degrees are all differentIf zz = -4 + j5, z2 = 3 - j2 , and z3 = 2 - j3 , what is Im (z; - 21) + 2; ? Note: 2* is the complex conjugate of z. Im (z) is the imaginary part of z O 1+ j2 O 11 + j2 none of the choices O -5+ j2