Math 514 homework #14 Due date: 19951013 D. J. Bernstein 19951006 A non-constant polynomial is reducible if it can be factored into two non-constant polynomials, irreducible otherwise. For example, x^{2} + 1 is reducible over C but irreducible over Q. Exercise 45. Let R be a Dedekind domain. Let F be the field of fractions of R. Prove “Gauss’s Lemma”: a monic polynomial f ∈ R[x] is irreducible over F if and only if it is irreducible over R. Exercise 46. Let R be a ring, let M be an R-module, and let t be an integer. Assume that len N ≤ t for every finitely generated R-submodule N of M. Prove that len M ≤ t.