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Name: Problem Set 1 Math 416, Section 200, Spring 2014 Due: Thursday, January 23. Review Sections 26, 27 in your textbook. Complete the following items, staple this page to the front of your work, and turn your assignment in at the beginning of class on Thursday, January 23. Remember to fully justify all your answers, and provide complete details. 1. Carefully read the entire course website. Send an email to your instructor containing: • Math416 in the subject line; • a review of the course website (What did you like? What would you change? List any typos you discovered, etc.); and • an acknowledgement that you understand the policies and procedures for this course. 2. Find all ideals N of Z45 . In each case, compute Z45 /N, that is, find a known ring isomorphic to the quotient ring. 3. Let R be a commutative ring with unity of prime characteristic p. Show that the map φ p : R → R given by φ p (a) = a p is a homomorphism (the Frobenius homomorphism). 4. Show that if N1 and N2 are ideals of R, then N1 ∩ N2 is also an ideal of R. 5. Let R be a commutative ring and N an ideal of R. Define the radical of N to be √ N B {x ∈ R | ∃n ∈ Z+ such that xn ∈ N}. √ Show that N is an ideal of R. 6. Exercise 37, p. 245. 7. Find all prime and maximal ideals of Z45 . 8. Let f (x), g(x) ∈ C[x]. a. Assume that f and g have no common zeros in C. Show that if f, g ∈ N for some ideal N of C[x], then N = C[x]. Hint: You may use the Fundamental Theorem of Algebra: Every non constant polynomial in C[x] has a zero in C. b. Show that if h(x) ∈ C[x] and h(α) = 0 for every common zero α ∈ C of f and g, then there exist n ∈ Z+ and q1 (x), q2 (x) ∈ C[x] such that (h(x))n = q1 (x) f (x) + q2 (x)g(x). Page 1 9. Exercise 34, p. 254. 10. Exercise 35, p. 254 11. Exercise 36, p. 254. Through the course of this assignment, I have followed the Aggie Code of Honor. An Aggie does not lie, cheat or steal or tolerate those who do. Signed: Page 2