Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 11 December 1, 2005

Problem 1: (a) Let R be an integral domain, a ∈ R^{∗} and b ∈ R. Prove that a polynomial f ∈ R[x] is irreducible in R[x] if and only if the polynomial f(ax + b) is irreducible in R[x].

(b) Let p be a prime. Prove that the polynomial f = x^{p−1} +x^{p−2} +⋯+x+1 is irreducible in Q[x]. (Hint: Consider f(x + 1) and use (a) and Eisenstein’s criterion.)

Problem 2: Prove that f = x^{4} + 1 is irreducible in Q[x]. (Hint: The following procedure works for all polynomial f Z[x]. If g|f one can assume deg g ≤ [\frac{deg f}{2} ] = s and one has g(a)|f(a) for all a ∈ Z. Choose different a_{0}, …, a_{s} ∈ Z with f(a_{i}) ≠ 0. For all s + 1-tupels (b_{0}, …, b_{s}) of integers such that b_{i}|f(a_{i}) choose the polynomial g ∈ Z[x] of degree ≤ s with g(a_{i}) = b_{i} and check if g|f. Use symmetries to simplify the computations.)

Problem 3: Let m and n be natural numbers. Determine the structure of the following four Z-modules:

(a) Hom_{Z}(Z, Z),

(b) Hom_{Z}(Z/mZ, Z),

(c) Hom_{Z}(Z, Z/nZ),

(d) Hom_{Z}(Z/mZ, Z/nZ).

Problem 4: List which of the 13 numbered Lemmas, Propositions, Thoerems and Corollaries in Section 3.1 remain true if one takes R-modules for arbitrary rings R with 1. Formulate modified versions if necessary.

Problem 5^{∗}: Find a counter-example to one of the numbered results of Section 3.1 for non commutative rings R.