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].

Step 1:Step 2:Step 3:Step 4:Step 5:Step 6:

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

Section: Chapter Questions

Problem 1RQ

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2. Let K be an extension of F and let a e K be algebraic over F. Then (i) There exists a unique monic irreducible polynomial p (x) e F (x] of least positive degree such that p (a) = 0. (ii) If g (x) e F[x] is such that g (a) = 0, then p (x) divides g (x).
Number 5
M3
2
3. Determine the Jordan Canonical Form of A if the characteristic polynomial of A is given by (1-A)²(2-x)³ with dim(Ker(A-I)) = 1 and dim(Ker(A- 21)) = 2.
{ (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 =
2. Suppose f is holomorphic in an open set UCC and a E U. Show that for any integer n > 0 there is a unique holomorphic function h:U → C such that f(z) = f(a)+f'(a)(z-a)+ f"(a)²- a)² 2! (z – a)" +f(m) (a). | +(z-a)"+'h(z), ze U. n! n+1
Let : R₁ → R₂ be a homomorphism of rings. Show that the mapping o': R₁[x] → R₂[x] given by o'(am +...+α₁x+αo): o(am) +...+ (a₁)x+ o(ao) is a homomorphism of rings with kero' = kero.
4. For each of the following polynomials p(r), find a number a such that p(x) is the minimal polynomial of a over Q. a) p(x) = x² + 2r-7 b) p(x)= x¹10x² + 1 -
5) Show that Z[i) ={a+ ib|a,b€Z} is an Euclidean Domain when we let the function %3D N: Z(6) \ {0}→ Z defined by N(a+ ib) = a + for all a, b EZ serve as the valuation.
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.
Number 6: show at K is splitting over Q.

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