Fall 2018

Math 5110 Graduate Algebra I

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Homework 4

Due Friday November 9, in class
  • Section 8.1
    • Homework: 2, 10, 12 and the following problem:
      • 13. Let \(D\) be a Euclidean domain with norm function \(N\). Suppose \(b \in D\) is neither zero nor a unit. Prove for every \(i \in \mathbb{Z} \cup \{0\}\) that \(N(b^i) < N(b^{i+1})\). (Source: Joseph J. Rotman, "Advanced Modern Algebra", Prentice Hall, Upper Saddle River (2002))
    • Hand In: 13
  • Section 9.3
    • Homework: 1, 3, 4 and the following problem:
      • 6. Let \(F\) be a field and let \(S\) be the set of all polynomials \(a_0 + a_1x +\ldots + a_nx^n\) in \(F[x]\) which satisfy \(a_0+a_1+\ldots+a_n=0\). Show that \(S\) is an ideal of \(F[x]\). Explain why \(S = (x-1)\). Prove that \(F[x]/S \cong F\). (Adapted from: Charles C. Pinter, "A Book of Abstract Algebra", Dover, Mineola, NY (2010))
    • Hand In: 6
  • Section 9.4
    • Homework: 1, 2, 5, 11 and the following problems:
      • 21. Prove that the following polynomials are irreducible in \(\mathbb{Z}[x]\).
        1. \(x^3+6x+12\).
        2. \(x^4+x^2+x+1\) (Hint: Show that the polynomial has no zeros mod 3 and that a factorization as a product of two quadratics forces impossible restrictions on the coefficients.)
        3. \(x^4+3x^3+2x^2+3x+4\).
        4. \(x^6+12x^5+49x^4+96x^3+99x^2+54x+15\) (Hint: use a suitable change of variables.)
        (Sources: Michael Artin, "Algebra", Prentice Hall, Englewood Cliffs, NJ (1991); Celine Carstensen, Benjamin Fine, Gerhard Rosenberger, "Abstract Algebra", De Gruyter, Berlin (2011); Joseph J. Rotman, "Advanced Modern Algebra", Prentice Hall, Upper Saddle River (2002); Derek J. S. Robinson, "Abstract Algebra (2nd edition)", De Gruyter, Berlin (2015) )
      • 22. Find a complete list of all polynomials over \(\mathbb{Z}_2\) of degree at most 4 that are irreducible over \(\mathbb{Z}_2\).
    • Hand In: 21, 22
  • Section 13.1
    • Homework: 1, 2, 3, 8
    • Hand In: 3