MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 12 Due Monday, September 7, 2009 (1) (a) Let R be a UFD (unique factorization domain, commutative), and let d a non-zero element in R . Prove that there are only ﬁnitely many principal ideals in R that contain d . (b) Give an example of a UFD R and a nonzero element d ∈ R such that there are in-ﬁnitely many ideals in R containing d . [No proof is required for this part; however, you must describe not only R and d , but also an inﬁnite family of ideals containing d .] (2) Suppose f ( x ) = 1 + x + x 2 + ··· + x p-1 , where p ∈ Z is prime. (a) Show that f is irreducible in Q [ x ]. [Hint: Write f ( x ) = ( x p-1) / ( x-1), and substitute x + 1 for x ]. (b) Show that ( p k ) = ∑ k +1 i =1 ( p-i p-k-1 ) for all k < p . (3) (a) All of the following rings R i , for i = 1 , . . . , 6 are additionally C-vector spaces. In each case, compute the vector space dimension by explicitly giving a basis for R i over C in each case. R
This is the end of the preview.
access the rest of the document.
Euclidean algorithm, Ri, Euclidean domain, Principal ideal domain, Matthew Macauley HW