= (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 (anxn + + a₁x + ao) = om (an)x+ + om(α₁)x+om (ao) is a ring homomorphism onto Zm[x]. (b) Show that if ƒ (x) ≤ Z[x] and Tm(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].
= (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 (anxn + + a₁x + ao) = om (an)x+ + om(α₁)x+om (ao) is a ring homomorphism onto Zm[x]. (b) Show that if ƒ (x) ≤ Z[x] and Tm(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].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(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) = om(an)x² +
is a ring homomorphism onto Zm[x].
+ om (a₁)x+om (ao)
=
(b) Show that if f(x) = Z[x] and ¯m (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 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%2F8084a024-ebf9-4fb7-8600-c7fa04d2572b%2Ff9a9d149-21fb-45d7-8ec8-360019b97506%2Fnvifr1j_processed.jpeg&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) = om(an)x² +
is a ring homomorphism onto Zm[x].
+ om (a₁)x+om (ao)
=
(b) Show that if f(x) = Z[x] and ¯m (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 Q[x].
(c) Use the previous part to show that x³ + 17x + 36 is irreducible
in Q[x].
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