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Math Help - Norms and common divisors

  1. #1
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    Norms and common divisors

    Define the norm N( \alpha) for an element  \alpha= a + b\sqrt{-5}   in Z\sqrt{-5} . Show that any common divisor of 1 + \sqrt{-5} and 2 in the ring Z\sqrt{-5} is a unit.

    I know that
    N( 1 + \sqrt{-5})= 6 and that
    N(2)= 4

    But how do you show that any common divisor is in the ring?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by lucyannepalmer View Post
    Define the norm N( \alpha) for an element  \alpha= a + b\sqrt{-5}   in Z\sqrt{-5} . Show that any common divisor of 1 + \sqrt{-5} and 2 in the ring Z\sqrt{-5} is a unit.

    I know that
    N( 1 + \sqrt{-5})= 6 and that
    N(2)= 4

    But how do you show that any common divisor is in the ring?
    You need to show that there exists no element with norm 2.

    Simples!

    (In general, this is made much easier when you use modulo arithmetic to eliminate big numbers.)
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  3. #3
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    Thanks for your quick reply.

    Why is it that you have to show there exists no element with norm 2? and how would I go about doing that?
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by lucyannepalmer View Post
    Thanks for your quick reply.

    Why is it that you have to show there exists no element with norm 2? and how would I go about doing that?
    Well, how is a norm defined? What is the property that it has?

    To prove that no element has norm 2, assume one exists. Assume there exists \alpha, \beta such that N(\alpha + \beta \sqrt{-5}) = 2, and just follow it through...
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  5. #5
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    Quote Originally Posted by lucyannepalmer View Post
    Define the norm N( \alpha) for an element  \alpha= a + b\sqrt{-5} in Z\sqrt{-5} . Show that any common divisor of 1 + \sqrt{-5} and 2 in the ring Z\sqrt{-5} is a unit.

    I know that
    N( 1 + \sqrt{-5})= 6 and that
    N(2)= 4

    But how do you show that any common divisor is in the ring?
    N(xy)=N(x)N(y) where x, y \in \mathbb{Z}[\sqrt{-5}].

    There are three things to consider:

    1. N(x) >= 0
    2. if N(x) = 1, then x is a unit.
    3. As Swlabr said, there is no element x such that N(x)=2.

    If the common divisor of your example, let's say k, has the norm N(k)=1, then k is the unit in your ring. Can you conclude now?
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  6. #6
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    So the only common divisor of 1 + \sqrt{-5} and 2, in the ring Z\sqrt{-5} is 1?
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  7. #7
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by lucyannepalmer View Post
    So the only common divisor of 1 + \sqrt{-5} and 2, in the ring Z\sqrt{-5} is 1?
    No. Any common divisor will have norm 1, and N(a) = 1 \Leftrightarrow a is a unit. This may be in your notes (the N(a) = 1 \Leftrightarrow a is a unit bit). Otherwise, it is good exercise to prove it.

    You need to prove that if a|2 and a| (1 + \sqrt{-5}) then N(a)=1. To do this, you must notice that as N(2) = N(1 + \sqrt{-5})=4 and so if a|2 and a| (1 + \sqrt{-5}) then N(a)|4. There are only three numbers which divide 4. They are 1, 2 and 4.

    If N(a)=1 then a is a unit.

    N(a)=2 cannot happen, you must prove this.

    If N(a)=4 then you need to prove that a cannot divide both ring elements. To do this, notice that there must exist b_1, b_2 such that ab_1 = 1 + \sqrt{-5} and ab_2 = 2 \Rightarrow N(ab_1) = N(ab_2) = 4 \Rightarrow N(b_1) = N(b_2) = 1 \Rightarrow 2 {b_{2}}^{-1}b_1 = 1 + \sqrt{-5}. Is this possible?
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  8. #8
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    Quote Originally Posted by lucyannepalmer View Post
    So the only common divisor of 1 + \sqrt{-5} and 2, in the ring Z\sqrt{-5} is 1?
    +1 and -1
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  9. #9
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    Thank you both, that was very helpful.
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