#3 on the picture b A MATLAB program that createsPOE Thomas W. Hungerford - Abstrac XO File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf+A' Read aloud V DrawF HighlightO Erase125of 6214.13 shows that p1(x)|q;(x) for somej. After rearranging and relabel-ing the q(x)'s if necessary, we may assume that pi(x)|q1(x). Since q1(x)is irreducible, p(x) must be either a constant or an associate of q(x).However, pi(x) is irreducible, and so it is not a constant. Therefore, pi(x)is an associate of q1(x), with p1(x) = q9(x) for some constant c1. Thus91(x)[c,P2(x)P;(x) • .• p(x)] = P(x)P{x) • .· p{x) = 9(x)4(x)•· · · q£x).Canceling q,(x) on each end, we haveP2(x)[cP3(x) • · • p,(x)] = 9{x)q;(x) •.• q&x).Complete the argument by adapting the proof of Theorem 1.8 to F[x],replacing statements about t, with statements about associates of9,(x). II ExercisesNOTE: F denotes a field and p a positive prime integer.A. 1. Find a monic associate of(a) 3x + 2x2 + x + 5 in Q[x](b) 3x – 4x2 + 1 in Zgx](c) ix +x – 1 in C[x]2. Prove that every nonzem f(x) E Fx] has a unique monic associate in F[x].3. List all associates of(a) x² + x + 1 in Z{x](b) 3x + 2 in Z,[x]4. Show that a nonzero polynomial in Z(x] has exactly p – 1 associates.5. Prove that f(x) and g(x) are associates in F[x] if and only if f(x)|g(x) andg(x)[f(x).6. Show that x +1 is irreducible in Q[x]. [Hint: If not, it must factor as(ax + b)(cx + d) with a, b, c, d e Q; show that this is impossible.]7. Prove that f(x) is irreducible in F[x] if and only if each of its associates isirreducible.Opte 2012Can lang A Rig Rat May aot be opd med or dptic e ae or in pat D so deais d trd party eonte may befnteBodk a Btalvtded thet ny pnd t d ot aty dha be oat ingpeias Ong Laing righto dddonl cot theit gta to in104 Chapter 4 Arithmatic in F[x]8. If f(x) E F[x] can be written as the product of two polynomials of lowerdegree, prove that f(x) is reducible in F[x]. (This is the second part of theproof of Theorem 4.11.)

Given that both are field hence we can Find corresponding unit of field then easily we can find its…

Answered: (e) +x-1 in Ca] 2. Prove that every…

(e) +x-1 in Ca] 2. Prove that every nonzem f(x) e Flx] has a unique monie asnciate in F1x. 3 List all asociater of (O P+x+1inZ{ 4. Show that a nonzero polynomial in Z has exactly p- 1 asociates.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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