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# Monthly Archives: August 2011

## 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
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## 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
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## UIC Master’s Exam- Fall 2007 R2

Problem Statement: is continuous on . Given that (a) prove has continuous derivative on (Hint: start with the change of variable u=x+t). (b) Given and show that there exists a such that for every . Solutions: (a)Proof: First let us start … Continue reading

## UIC Master’s Exam- Fall 2007 R1

Problem Statement: Consider the power series (a) For which values of does it converge absolutely? Conditionally? (b) Show that on the convergence is uniform. Solutions: (a) Claim: the series converges absolutely on and conditionally on . Consider , then it follows that … Continue reading

## 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
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## 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
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## 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
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