Thank you for your help I need a detailed solution of this p

Thank you for your help. I need a detailed solution of this problem ASAP. (AMS351 Book : Numbers, Groups & Codes six edition by Hemphreys & Prest)

Exercises 6.3 1. Prove, by induction on r, that if f is an irreducible polynomial and f divides the product fi ....fr, then f divides one of fi, f2, fr.

Solution

Let p, t,s F[x],where F is a field and let p = rs be an irreducible polynomial over the field F such that p does not divide t. If p|rs, then ts = kp for some k F[x]. Then gcd(p,t) must be a divisor of the irreducible polynomial p, so that gcd(p,t) is either 1 or the unique monic associate of p. However, gcd(p,t) cannot be the unique monic associate of p as p is not a divisor of t. Hence gcd(p,t)= 1. Therefore, 1 = ap+bt for some a,b F[x]. Now, on multiplying both the sides by s, we get s = (ap+bt)s = aps+bts = aps +bkp =p(as+bk).This implies that p|s. Then, by induction, if   f an irreducible polynomial in F[x] and f divides the product f₁…f_r, then f divides one of f₁ ,f₂,…, f_r.