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Math Help - Rings: Principal Ideal domain

  1. #1
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    Rings: Principal Ideal domain

    Hey Guys,

    I've never been on one of these sites before but am really stuck on an problem sheet I have been set. I'm doing fine in all my other modules but this algebra one doesn't seem to click well with me
    Any help would be much appreciated.

    So I have to prove that R is a principal ideal
    domain where R is the ring {a + b(sqrt(-2)) | a,b are integers}

    (by b(sqrt(-2)) I mean b multiplied by the square root of -2)

    TY in advance

    x
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  2. #2
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    Quote Originally Posted by sarah89 View Post
    Hey Guys,

    I've never been on one of these sites before but am really stuck on an problem sheet I have been set. I'm doing fine in all my other modules but this algebra one doesn't seem to click well with me
    Any help would be much appreciated.

    So I have to prove that R is a principal ideal
    domain where R is the ring {a + b(sqrt(-2)) | a,b are integers}

    (by b(sqrt(-2)) I mean b multiplied by the square root of -2)

    TY in advance

    x
    define the map f: R \setminus \{0 \} \longrightarrow \mathbb{N} by f(a+b \sqrt{-2})=a^2+2b^2. show that f is a norm-Euclidean and so R is an Euclidean domain and therefore a PID.
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