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### Homework 4

- 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

- Homework: 2, 10, 12 and the following problem:
- 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

- Homework: 1, 3, 4 and the following problem:
- Section 9.4
- Homework: 1, 2, 5, 11 and the following problems:
- 21. Prove that the following polynomials are irreducible in \(\mathbb{Z}[x]\).
- \(x^3+6x+12\).
- \(x^4+x^2+x+1\) (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.)
- \(x^4+3x^3+2x^2+3x+4\).
- \(x^6+12x^5+49x^4+96x^3+99x^2+54x+15\) (Hint: use a suitable change of variables.)

- 22. Find a complete list of all polynomials over \(\mathbb{Z}_2\) of degree at most 4 that are irreducible over \(\mathbb{Z}_2\).

- 21. Prove that the following polynomials are irreducible in \(\mathbb{Z}[x]\).
- Hand In: 21, 22

- Homework: 1, 2, 5, 11 and the following problems:
- 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