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Homework Help: Need help deciphering this number theory problem

  1. Apr 30, 2014 #1
    1. The problem statement, all variables and given/known data

    What does triangle line mean? What is "+" for sets here?

    Once I know that, if I need assistance, I will show an attempt. Otherwise I will be satisfied. :)

    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

    Last edited: Apr 30, 2014
  2. jcsd
  3. Apr 30, 2014 #2
    The triangle means "is ideal of". So ##K_1## and ##K_2## are ideals of ##R##. We then define

    [tex]K_1 + K_2 = \{a+b~\vert~a\in K_1,~b\in K_2\}[/tex]

    This is again an ideal of the ring.
     
  4. Apr 30, 2014 #3
    Ok, and while we're at it, what on earth does equivalence mod "an ideal" mean? Does it mean the full set of equivalences for all elements in an ideal?

    I am referring to the statement "x =K1 a." I just realized the picture is sideways.

    And why is it stating that "if the following system is solvable.." since when is a system of congruences with coefficients of 1 ever not solvable?

    Thank you.
     
  5. Apr 30, 2014 #4
    It means ##x-a\in K_1##.

    For example, the system ##x\equiv_2 0##, ##x\equiv_2 1## is not solvable. This is equivalence modulo the ideal ##2\mathbb{Z}##.
     
  6. Apr 30, 2014 #5
    Alrighty, thanks for filling in the gaps for me, hah!
     
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