## Math 5110 Graduate Algebra I

### 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 Euclidean valuation function $$\nu$$. Suppose $$b \in D$$ is neither zero nor a unit. Prove for every $$i \in \mathbb{Z} \cup \{0\}$$ that $$\nu(b^i) < \nu(b^{i+1})$$. (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
• Section 13.2
• Homework: 1, 2, 3, 7, 8, 11, 19-21
• Hand In: 3, 8, 19-21