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Thread: Gauss Lemma (Number Theory)

  1. #1
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    Gauss Lemma (Number Theory)

    Use Gauss Lemma (Number theory) to calculate the Legendre Symbol $\displaystyle (\frac{6}{13})$.

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

    $\displaystyle (\frac{a}{p})=(-1)^n$. They say: Let $\displaystyle \pm m_t$ be the least residue of $\displaystyle ta$, where $\displaystyle m_t$ is positive. As $\displaystyle t$ ranges between 1 and $\displaystyle \frac{(p-1)}{2}$, 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
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    Quote Originally Posted by mathsss2 View Post
    Use Gauss Lemma (Number theory) to calculate the Legendre Symbol $\displaystyle (\frac{6}{13})$.

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

    $\displaystyle (\frac{a}{p})=(-1)^n$. They say: Let $\displaystyle \pm m_t$ be the least residue of $\displaystyle ta$, where $\displaystyle m_t$ is positive. As $\displaystyle t$ ranges between 1 and $\displaystyle \frac{(p-1)}{2}$, 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
    In $\displaystyle \{ 6,12,18,24,30,36\}$ the corresponding remainders are $\displaystyle \{6, -1, 5, -2, 4, -3 \}$.
    There are $\displaystyle 3$ negative signs.
    Therefore, $\displaystyle (6/13) = (-1)^3 = -1$.
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