= (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].
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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![(19) Let om: Z → Zm be the ring homomorphism defined by om(a)
remainder of division of a by m.
(a) Show that m : Z[x] → Zm[x] defined by
om (anx" +...+ a₁x + ao)
+ a₁x + ao) = om(an)x² + + om(a₁)x+ om (ao)
is a ring homomorphism onto Zm[x].
=
(b) Show that if ƒ(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree
n and om(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].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba0acd94-ebc3-41e6-80c8-3fde2b12e743%2F520b7075-afba-4700-bc1c-50c301a4602a%2Ffu9ejo_processed.png&w=3840&q=75)
Transcribed Image Text:(19) Let om: Z → Zm be the ring homomorphism defined by om(a)
remainder of division of a by m.
(a) Show that m : Z[x] → Zm[x] defined by
om (anx" +...+ a₁x + ao)
+ a₁x + ao) = om(an)x² + + om(a₁)x+ om (ao)
is a ring homomorphism onto Zm[x].
=
(b) Show that if ƒ(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree
n and om(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].
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