F. Mapping onto z to Determine Irreducibility over o If h :z z, is the natural h

Question

F. Mapping onto z to Determine Irreducibility over o If h :z z, is the natural homomorphism, let h : Z [x] Zn[x] be defined by (ao + aix + + a"x") = h(ao) + h(a)s + + h(an)xn In Chapter 24, Exercise G, it is proved that & is a homomorphism. Assume this fact and prove: #1 If6(a(x)) is irreducible inz,[x] and a(x) is monic, then a(x) is irreducible in z(x]. 210+7 is irreducible in ofx] by using the natural homomorphism from z to 2s 3 The following are irrducible in oll (find the right value of n and use the natural homomorphism from z to zn):

Explanation / Answer

Polynomials behave similar to Z .As for the integers Z, we can define gcd and lcm for polynomials. There also exists an Euclidean algorithm To prove these results for polynomials, one could simply copy the proofs for the same results for the integers. Suppose that f(x), g(x) K[x]. First, we say that a polynomial g divides a polynomial f if f(x) = g(x)h(x) for some polynomial h K[x]. A monic polynomial f is called irreducible if it has exactly 2 monic divisors

By the above statement it can be said that a(x) is irreducible in z[x] where f(x) = g(x)h(x) is taken as example for reference.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.