= (19) Let om : Z → Zm be the ring homomorphism defined by σm(a) = remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + +a12 + ao) = om (an) x + + om(a₁)x+om (ao) is a ring homomorphism onto Zm[x]. (b) Show that if ƒ(x) = Z[x] and σm(f(x)) = Zm[x] both have degree n and σm (f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible in Q[x]. (c) Use the previous part to show that x³ + 17x +36 is irreducible in Q[x].

= (19) Let om : Z → Zm be the ring homomorphism defined by σm(a) = remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + +a12 + ao) = om (an) x + + om(a₁)x+om (ao) is a ring homomorphism onto Zm[x]. (b) Show that if ƒ(x) = Z[x] and σm(f(x)) = Zm[x] both have degree n and σm (f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible in Q[x]. (c) Use the previous part to show that x³ + 17x +36 is irreducible in Q[x].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 10E: Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove...

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