18.1.26. Define an onto homomorphism f: (Z/36Z) [x] →Z/36Z such that ker(ƒ)=(x).(a) Is (x) prime and/or maximal in (Z/36Z)[x]?(b) Let AB ==(6).(3) be an ideal of Z/36Z. Find f¯¹(A). Do the same for(c) Find a familiar ring that is isomorphic to (Z/36Z)[x]/ƒ¯¹(A). Dothe same for f¹(A)/(x), and ƒ¯¹(B)/(x).(d) Find a ring of the form (Z/36Z)/?? that is isomorphic to (Z/36Z)[x]/f-¹(B).(e) Find two maximal ideals in (Z/36Z) [x].

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Answered: 18.1.26. Define an onto homomorphism f:…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Suppose L: R2 R2 is defined by L →→ ordered bases. [L(u)]B₁ = [L(u)]B₂³ = · 4 [X₂] - [×₂ -³x, ] · Let B, = {[§ ]· [²]} and B₂ = {[2]· [4]} = X2 1 be ordered bases for R². If u = [₁2], find [L(u)]g, and [L(u)]g,, the coordinate vectors of L with respect to each of the given 12
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19.2.8. Let F7 = Z/7Z be the field with seven elements, and let f,g Є F7[x] be defined by f(x) = x³ + x³ + x + 1 and g(x) = x² + x. Is f in the ideal generated by g? If the answer is no, then find h = F7[×] with degree less than two such that f − h is in the ideal generated by g.
{ (80) : a, b,ce z}. and for all such A E L let f, g, h: L→ Z be given by b f(A) = a, g(A) = tr(A), h(A) = det(A) (a) Show that f is a surjective homomorphism. (b) Show that g is additive, but not multiplicative. (c) Show that h is multiplicative, but not additive. 7. Let L =
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1. Find curl(v) of the following in terms of (x1, x2, x3) and (e1, e2, e3). If curl(v) = 0, find the function f such that Vf = v. a. v(x) = (x² + x + x3)-'(x1e1 + x2€2 + X3@3). b. v(x) = (xỉ + X2X3)e1 + (x3 + X1X3)e2 + (xž + x2X1)e3 c. v(x) = e"1(sin(x2) cos(x3)e1 + sin(x2) sin(x3)e2 + cos(x2)e3)
5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0) = p(1) = p'(1) = 0. The dimension of W is (a) 1 (b) 7 (c) 8 (d) 9 (е) 11
10. Consider the irreducible polynomial f = X' + X³ + X² + X + 1 in Z[X]. Let a = [X] in the ring Z[X]/(f). Express (a + 1)(a² + 1) in the form aza + aza² + aa + ao where a; E Z.
d) Draw an arrow diagram of the inverse for the bijection function found in (c). Define W:Z → Z and V: Z → Z by the rules W(a) = a? and V (a) = a mod 5 for all integers a. Find the following: e) (W • V)(6) (V • W)(9) iii. i. ii. Is (W • V) = (V •W) for all integers a e Z? Justify your answer. -1-
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Let H ={[1 a] : a in Z} {[0 1] Define a map φ : Z → H by φ(n) = [1 n] for all n in Z. [o 1] a. Show that φ is one-to-one.(Definition of 1 − 1: If φ(x) = φ(y), then x = y. To prove this, Assume φ(x) = φ(y) andshow that x = y.) b. ) Show that φ is onto.(Definition of onto: For every y ∈ H, there is x ∈ Z such that φ(x) = y. To prove this, startwith a matrix in H and then find an element in Z that is mapped to that matrix.) c. Show that φ is operation preserving.(Show: φ(x+y) = φ(x)φ(y). To do this, compute φ(x+y) and then computer φ(x)φ(y) andcompare them.) d Is H ≈ Z? Explain.
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