4. In Z[z], let be the ideal generated by z and 4 (that is, the set ofall elements of the form 4a + zg(x))(a) Show 14(b) Prove < 2,4> is not a principal ideal, that is, #for any h(z) E Z[x]

Given: x,4 is an ideal generated by x and 4. To show : 1∉x,4 and x,4 is not a principal ideal.

Answered: 4. In 2[z], let < 2,4> be the ideal…

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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(1) Let I be a proper ideal of the commutative ring R with identity. Then I is a ........ if and only if the quotient ring R/I is a field. (i) prime ideal (ii) primary ideal (iii) Maximal ideal (2) Z/(5): (i) (5) (ii) 5Z (iii) {[0],[1],[2],[3],[4]} (3) If R is an integral domain has non zero characteristic, then Char(R)=..... (i) 5 (ii) 4 (iii)9 (4) Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to ... (i)I+J (ii)I.J (iii) I.J+ I (5) If f(x) then f(x) is reducible in Z5[x]. (i) x³ + 2x² + 2x + 1 (ii) x² + 1 (iii) x² + 2
2
In Z{x}, Let I = {f(x) ∈ Z[x] / f(0) is an even integer} Prove that I = <x,2> is I prime ideal of Z{x}? is I a maximal ideal of Z{x}? How many elements does Z{x}/I have?
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.
In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer}. 1) Prove that I is an ideal of Z[x]. 2) What is the elements of Z[x]/1?
(a) Find all solutions of the equation x³+x²-2x=0 in Z₁. (b) Let R be a ring. Define what it means for R to be an integral domain.
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.
If R= Z[x], f(x) = x2 +1, and g(x) = a0+a1x+...+anxn is a polynomial of degree 2 or more in R. How do you prove that g(x) is congruent to g(x)-anxn-anxn-2 mod I? I is the principal ideal generated by f(x)
This question is of Ring theory
(9) Let f(r) be an Irreducible cubic over Q with cyclic Galols group. Show that all roots of f(z) are real.
In the notation of Example 7, show that d(xy) = d(x)d(y).
(7) In the ring (Z, +,.), we get n {P:P non trival prime ideal in Z} = ...... (а) ф (b) (Z, +, .) (c) ({0}, +,.) (d) No Choice (a) (b) (c) (d)

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4. In 2[z], let < 2,4> be the ideal generated by z and 4 (that is, the set of all elements of the form 4a + zg(x)) (a) Show 1 <1,4> (b) Prove < 2,4> is not a principal ideal, that is, <1,4> for any h(z) E Z[x]