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Gauss Lemma (Number Theory)

  1. Nov 3, 2008 #1
    Use Gauss Lemma (Number theory) to calculate the Legendre Symbol [tex](\frac{6}{13})[/tex].

    I know how to use Gauss Lemma. However we use the book: Ireland and Rosen. They define Gauss Lemma as:

    [tex](\frac{a}{p})=(-1)^n[/tex]. They say: Let [tex]\pm m_t[/tex] be the least residue of [tex]ta[/tex], where [tex]m_t[/tex] is positive. As [tex]t[/tex] ranges between 1 and [tex]\frac{(p-1)}{2}[/tex], n is the number of minus signs that occur in this way. I don't understand how to use this form of Gauss's Lemma
     
    Last edited: Nov 3, 2008
  2. jcsd
  3. Nov 3, 2008 #2

    gabbagabbahey

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    What are [itex]a[/itex] and [itex]p[/itex] in this case? What does that make [itex]\frac{(p-1)}{2}[/itex] ? What does that make the least residue of [itex]ta[/itex] in this case?
     
  4. Nov 3, 2008 #3
    Could you be more specific, I really do not know how to use this version of Gauss's Lemma. Could you show me some steps on how to start it this way?
     
  5. Nov 3, 2008 #4

    gabbagabbahey

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    You want to use the lemma for [itex]\left( \frac{6}{13} \right)[/itex], which means you want an "a" and "p" such that [itex]\left( \frac{a}{p} \right) = \left( \frac{6}{13} \right)[/itex] where "p" is a prime....surely you can think of at least one "a" and one "p" for which this will hold true?
     
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