Problem 2(a) Let f: R →S be an onto homomorphism of rings, Let I be an ideal of R.Show that f(1)a)laEI) is an ideal of S (check additive subgroup and ideal condition).(b) Recall the substitution homomorphism 4 : Q[x]→Q[v2]1>given by 4a (p(x)) = p(V2)You can assume this is a homomorphism.(i) Show pz is onto.(ii) Express Ker uz as a principal ideal of Q[x] justify).(iii) What conclusion can be drawn using FHT (the Fundamental Homomorphism Theorem)?

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Answered: Problem 2 Show that f () = @)| a e 1)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let a e Zz be a zero of the irreducible polynomial p(x) = x³ + x² + 2 in Z3[x]. a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of an element in Z3(a). b. Show that 2a2 +1 is also a zero of p(r). = 3 for all BE Z3(a) \ Z3. c. Explain why deg(B, Z3) d. Determine irr(2a, Z3).
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1 2 3 4 5 6 3 6 4 51 2 1 2 3 4 56 2 4 6 5 3 1)! Consider permutations o = and r = in S6. Find: (a) composition too; (b) a-. (b) Determine the domain for each of the following functions. 1 1.) f(2)- 1 4.) m (x) = V – 6x 7.) f(x) + Vr? - 4x + 3 I+5 12 – 25 I +7 1² +1 1. 2.) g(x) = VI + 5 5.) f (1) = 8.) g(z) = Vr? + 4- %3D %3D %3D Va +2–3 3.) p(z) 6.) t(r). 9.) f(x) = Vr -1+ v0 %3D 2 – 6x
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Problem 2 Show that f () = @)| a e 1) is an idenl of S (eheck additive subgroup and ideal condition). hinm u a: Q[x] Q[V2]