{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Homework1 - Homework 1 Solutions 1 Case 2 of the Division...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Homework 1 Solutions 1. Case 2 of the Division Algorithm follows almost identically to Case 1. The big change is replacing q + 1 in the existence proof with q - 1 since a is now negative. 2. Problem 1.51: Let I = { k Z : ζ k = 1 } ⊆ Z . We claim that I meets the hypotheses of Corollary 1.37. (1) follows clearly since ζ 0 = 1. (2): Suppose a, b I then ζ a - b = ζ a ζ b = 1 1 = 1 which implies a - b I . (3): Suppose a I and q Z . Then ζ qa = ( ζ a ) q = 1 q = 1 which implies qa I . Since I meets the hypotheses of Corollary 1.37 we know that there exists a nonnegative integer d such that I consists precisely of all the multiples of d . Furthermore, d 6 = 0 since ζ is a root of unity and thus k 6 = 0 such that ζ k = 1. Hence we have a positive d as desired. 3. Problem 1.56: Let a, b Z such that as + bt = 1 for some s, t Z . From the proof of Theorem 1.35 we know that ( a, b ) is the smallest positive linear combination of a and b . Since as + bt = 1, 1 must be the smallest positive linear combination (there are no smaller positive integers). Therefore (
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern