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Posers and Puzzles

Posers and Puzzles

  1. Standard member Fiathahel
    Artist in Drawing
    03 Feb '04 11:10
    Find all elements in Q(sqrt(5)) which are a root of a polynomial with integer coefficient (and post them in this thread ).

    Q(sqrt(5)) is the smallest field containing the rationals and sqrt(5).

    Steffin
  2. Standard member TheMaster37
    Kupikupopo!
    03 Feb '04 13:11
    LOL

    And if that's too easy, try the same for Q(sqrt( -10 + sqrt (10)) ).
  3. Donation Acolyte
    Now With Added BA
    06 Feb '04 11:42
    Originally posted by Fiathahel
    Find all elements in Q(sqrt(5)) which are a root of a polynomial with integer coefficient (and post them in this thread ).

    Q(sqrt(5)) is the smallest field containing the rationals and sqrt(5).

    Steffin
    All of them are algebraic. What are you getting at?
  4. Standard member Fiathahel
    Artist in Drawing
    06 Feb '04 12:12
    Originally posted by Acolyte
    All of them are algebraic. What are you getting at?
    Of course they are algebraic. But the task is to find them.
  5. Donation Acolyte
    Now With Added BA
    06 Feb '04 15:56
    Originally posted by Fiathahel
    Of course they are algebraic. But the task is to find them.
    Q(sqrt(5)) is all numbers of the form a + b*sqrt(5), where a and b are rationals. Do you mean produce polynomials that have these numbers as roots? Well, p/q + (r/s)*sqrt(5) is a root of the following:

    f(x) = QSX - 2pqSx + (PS - 5RQ) = 0

    where P, Q etc mean p^2, q^2 etc.
  6. Standard member Fiathahel
    Artist in Drawing
    06 Feb '04 16:34 / 1 edit
    Originally posted by Acolyte
    Q(sqrt(5)) is all numbers of the form a + b*sqrt(5), where a and b are rationals. Do you mean produce polynomials that have these numbers as roots? Well, p/q + (r/s)*sqrt(5) is a root of the following:

    f(x) = QSX - 2pqSx + (PS - 5RQ) = 0

    where P, Q etc mean p^2, q^2 etc.
    Sorry Acolyte, I forgot something. You were right all of them met the criteria. I forgot to say that the leading coefficient of the polynomial had to be 1.
    So again:

    Find all elements in Q(sqrt(5)) which are a root of a polynomial with integer coefficient and leading coefficient 1