Math 514 homework #7 Due date: 19950918 D. J. Bernstein 19950911 CANCELLATION: NO CLASS FRIDAY, 15 SEPTEMBER. We’re holding a number theory conference in honor of Professor Atkin, who has just retired. There is a good lineup of speakers, and you may enjoy some of the talks; I don’t think much background will be required for the 10:15 and 11:30 talks on Thursday and Friday. The Thursday and Saturday talks are in Lecture Center F4. The Friday talks are in Chicago Rooms A and B, Illini Union, 818 S. Wolcott, on the west campus. You can find a complete conference schedule on the SEO third floor seminar board. New approach to timing: Class will begin when my watch says 12:00. Exercise 19. Find a monic integer polynomial with \sqrt{2} + ^{3} \sqrt{3} as a root. The following two exercises take place in a ring R. Recall that a and b are coprime in R if ax + by = 1 for some x, y ∈ R. Exercise 20. Prove that if a divides bc, with a and b coprime, then a divides c. Exercise 21. Prove the “rational root theorem”: if f_{d}a^{d} + f_{d−1}a^{d−1}b + ⋯ + f_{1}ab^{d−1} + f_{0}b^{d} = 0, with a and b coprime, then a divides f_{0} and b divides f_{d}. A polynomial f_{d}z^{d} + f_{d−1}z^{d−1} + ⋯ + f_{0} is monic if f_{d} = 1. Exercise 22. Prove that Z is integrally closed: if x ∈ Q is a root of a monic integer polynomial then x ∈ Z. (Hint: Any rational number may be written as the quotient of two coprime integers.)