# Gauss Lemma (Number Theory)

• Nov 2nd 2008, 10:55 PM
mathsss2
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
• Nov 3rd 2008, 08:05 PM
ThePerfectHacker
Quote:

Originally Posted by mathsss2
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, $\displaystyle (6/13) = (-1)^3 = -1$.