Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Instructor: Roy Joshua, Office MW 604. E-mail: joshua@ math.ohio-state.edu
We will try to adopt the following schedule: Lectures, MWF 1 hour each (instead of 48 minutes) Recitation by T
Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework V: Field and Ring Theory
1. Let F be a field with n elements and let f (x), g(x) F [x] so that f (x) = g(x) and f (a) = g(a) for all a F . Show that deg(f (x) - g(x) n. 2. Let
Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework IV: Ring Theory
1. Show that Z[i] is a Euclidean domain with (x) = a2 + b2 if x = a + bi. Show that 3 Z[ 5i] is irreducible, but not prime. 2. Find a prime ideal of Z16 [x] th
Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework III: Ring Theory
1. Let A be a not-necessarily commutative integral domain with 1. Suppose that a, b A are such that ab = 1. Show that then ba = 1 and hence that a and b are un
Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework II: Group Theory
1. Let G be a non-abelian group of order p3 for some prime p. Show that Z(G) = G . 2. Let G be a finite group and P a Sylow subgroup. Show that NG (NG (P ) = N
Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework I: Group Theory
1. Let f : G H be a bijective homomorphism of groups. Show that f -1 is also a homomorphism. 2. Let G be a group of order 2p for some odd prime p. Show that G