# Gauss Lemma (Number Theory)

1. Nov 3, 2008

### mathsss2

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

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

$$(\frac{a}{p})=(-1)^n$$. They say: Let $$\pm m_t$$ be the least residue of $$ta$$, where $$m_t$$ is positive. As $$t$$ ranges between 1 and $$\frac{(p-1)}{2}$$, 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

Last edited: Nov 3, 2008
2. Nov 3, 2008

### gabbagabbahey

What are $a$ and $p$ in this case? What does that make $\frac{(p-1)}{2}$ ? What does that make the least residue of $ta$ in this case?

3. Nov 3, 2008

### mathsss2

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?

4. Nov 3, 2008

### gabbagabbahey

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