#19 from book b My Questions | bartlebyThomas W. Hungerford - Abstrac ×+O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdfA' Read aloud | V DrawF HighlightO Erase142of 621T2. Let F be a neid and jX) E FXI. II CE Fand JX + C) IS Irreaucibie in F|X],prove that f(x) is irreducible in F[x]. [Hint: Prove the contrapositive.]13. Prove that f(x) = xt + 4x + 1 is irreducible in Q[x] by using Eisenstein'sCriterion to show that f(x + 1) is irreducible and applying Exercise 12.14. Prove that f(x) = x* + x + + x + l is irreducible in O[x). [Hint: Use thehint for Exercise 21 with p = 5.]15. Let f(x) = a,t + an-1x1 +• ·•+ a,x + a, be a polynomial with integercoefficients. If p is a prime such that p|a1, p|az, ... ,P|a, but p } m, andOt 2012 C Le A Rig taved May at be opind cd ar we ar la pt Dete drie d perty cot y bepd me Bootendtr C o. Edalvew badd t oy ppd t dan at ty het he ovlrgpert Cgge Laig ma right tomveddonl at y tme i dghs anmguire120 Chapter 4 Arithmetic in F[x]pta, prove that f(x) is irreducible in Q[x). [Hint: Let y = 1/x in f(x)/x"; theresulting polynomial is irreducible, by Thcorem 4.24.]16. Show by example that this statement is false: If f(x) E 7[x] and there is noprime p satisfying the hypotheses of Theorem 4.24, then f(x) is reducible in Qx].17. Show that there are n+1 - n polynomials of degree k in Z[x].18. Which of these polynomials are irreducible in Q[x]:(a) メ-+1(c) x + 4x + 2x+ 3x - x + 5(b) x* +x +1(d) x + 5x + 4x + 719. Write each polynomial as a product of irreducible polynomials in Q[x].(a) + 2x – 6x? – 16x – 8() x" – 2x – 6x– 15x – 33x – 920. If f(x) = a, +..+ a,x + ao, 8(x) = b,x + ...+ b,x + bo, and h(x) =cxt + ...+ cx + co are polynomials in Z[x] such that f(x) = g(x)h(x), showthat in Z[x], F(x) = (x). Also, see Exercise 19 in Section 4.1.C.21. Prove that for p prime, f(x) = x-1+ -2+ . ..+ x² + x + 1 is irreduciblein Q[x]. [Hint: (x – 1)f(x) = x – 1, so that f(x) = (x – 1)/(x – 1) andf(x + 1) = [(x + 1y –11/x. Expand (x + 1)® by the Binomial TheoremCoursSa(Appendix E) and note that p divideswhen k > 0. Use Eisenstein'sCriterion to show that (x + 1) is irreducible; apply Exercise 12.]EXCURSION: Geometric Constructions (Chapter 15) may be covered atthis point if desired.4.6Irreducibility in R[x] and C[x]*3:03 PMO Search for anything口EPICAiへEPIC11/20/2020

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