absract algebra 1. Consider the ring Z[x]. Prove that the ideal (2, r) = {2f(x)+xg(x) : f(x), g(x) E Z[]}is not a principal ideal; that is, show that (2, r) # (p(x)) for any p(x) E Z[r].2a

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Answered: 1. Consider the ring Z[r]. Prove that…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.2: Divisibility And Greatest Common Divisor

Problem 34E

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2. Prove that I := {f(x) = R[x] : f(1) = 0} is an ideal of R[x] and find a well-known ring R₂ such that R[x]/IR₂ as rings.
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3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[x]/I provides an abstract construction of the ring R[ɛ]. (a) Define a function : R[ɛ] → R[x]/I by (a + b) = a + bx + I Verify that is a ring homomorphism. (b) Consider a polynomial f(x) = a + bx + c2x² + €3x³ + ….. +Cnx² € R[x]. Show that f(x) + I = a + bx + I are equal as equivalence classes. (c) Using part (b), explain why the homomorphism from part (a) is surjective.
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.
Show that over the ring Z/30Z[x] the polynomial f(x)=x has the factorization f(x)=(10x+21)(15x+16)(6x+25)
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Consider the quotient ring R1 = Z3[x]/(x² + 1) and R2 = Z3[i] = {a + bi / a, b E Z3} with operations (a + bi) + (a' + b'i) = (a + a') + (b +b')i, (a + bi) · (a' + b'i) = (aa' – bb') + (ab' + a'b)i. (a) Show an isomorphism o : Rị → R2 between these rings. (Explain the operations in R1 and how ø respects them). (b) Show that x2 +1 is irreducible in Z3[]. So R1 - R2 is a field. (c) Find (1+ 2i)-1 in this field.
3
B. If R = Z[x] is the polynomial ring over Z, decide whether the following ideals are maximal ideals of R and justify your reasoning: (i) I = (3,x²+1); (ii) I=(3,x³+x+1).

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1. Consider the ring Z[r]. Prove that the ideal (2, x) = {2f(x)+rg(x) : f(x), g(x) E Z[r]} is not a principal ideal; that is, show that (2, r) (p(x)) for any p(x) E Z[r].