Category Archives: Math

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

Posted in Analysis, Continuity, Differentiable, Fundamental Theorem of Calculus, Math | Leave a comment

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

Posted in Absolute Convergence, Analysis, Conditional Convergence, M-Test, Math, Series, Series of Functions, Uniform Convergence | 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. 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

Gallian: Ch. 3 #35

Problem Statement: Prove that a group of even order must have an element of order . Proof: Let be a group such that and consider the set . is the set of all elements of that are not of order . … Continue reading

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

Rudin: Ch. 5 #4

Problem Statement: If where are real constants, prove that has at least one real root between and . Proof: Consider the polynomial on . Then it follows that and which equals by our assumption. Furthermore, since is a polynomial with real coefficients … Continue reading

Posted in Analysis, Differentiable, Math, MVT | Leave a comment