Category Archives: Algebra

Suzie-Q

Problem Statement: Let be an integral domain. Show that is a field if and only if has no nontrivial ideals. Proof: First assume that is a field and let be an ideal of . If then it follows that since for every … Continue reading

Posted in Algebra, Field, Homomorphism, Ideal, Integral Domain | 1 Comment

UIC Master’s Exam- Spring 2007 A2

Problem Statement: (a) State the division algorithm for (b) Show that is a principle ideal domain. Solutions: (a) The Division Algorithm: If with then there exist such that where . (b) Proof: Let be an ideal of . If is the trivial ideal … Continue reading

Posted in Algebra, Principal Ideal, Principal Ideal Domain | Leave a comment

UIC Master’s Exam- Fall 2007 A3

Problem Statement: Show that a finite integral domain is a field. Proof: Fix  and consider the function for all . We wish to show that is injective, and thus has an inverse. Suppose that , then which implies that . Since … Continue reading

Posted in Algebra, Field, Integral Domain, Math | 1 Comment

UIC Master’s Exam- Fall 2007 A2

Jeremy is preparing for the University of Illinois at Chicago Master’s Exam this coming Tuesday and so we have been studying together. His exam covers many topics, including algebra and analysis, and so you will be seeing several posts from … Continue reading

Posted in Algebra, Field, Integral Domain, Math, Principal Ideal | Leave a comment

Gallian- Ch.9 #49

Problem Statement: If , prove that . Proof: Recall that is the set of all elements of which commute with all elements of . Let , we wish to show that . So consider an element . Then there is some such … Continue reading

Posted in Algebra, Centralizer, Normal | Leave a comment

The G/Z Theorem

Theorem: Let be a group and let be the center of . If is cyclic, then is Abelian. Proof: Suppose that is cyclic. Then there is some such that . Let . We wish to show that . It follows that there … Continue reading

Posted in Abelian, Algebra, Center, Cyclic, Group | Leave a comment

Gallian- Ch. 8 #11

Problem Statement: How many elements of order does have? Explain why has the same number of elements of order as . Generalize to the case of . Solution: First let’s find how many elements there are of order in . An element has … Continue reading

Posted in Algebra, Cyclic, Group, Math, Order | Leave a comment