Find the value of the following Legendre Symbols and Use Gausslemma to compute each of the Legendre symbols below

Question

Find the value of the following Legendre Symbols:

a) $(\frac{18}{43})$

b) $(\frac{19}{23})$

Use Gauss' Lemma to compute each of the Legendre symbols below:

a) $(\frac{8}{11})$

b) $(\frac{6}{31})$

Answer 1

Hint

$$a)\;\;\left(\frac{18}{43}\right)=\left(\frac2{43}\right)\left(\frac9{43}\right)=(-1)\cdot1=-1$$

Explanation:

$$\left(\frac2p\right)=1\iff p=\pm1\pmod 8\;,\;\;p\;\;\text{a prime}$$

You do now the second one and as for the second part, for example

$$\left\{\;8\cdot 1=8\;,\;\;8\cdot2=5\;,\;\;8\cdot3=2\;,\;\;8\cdot4=10\;,\;\;8\cdot5=7\;\right\}\;,\;\;\text{everything}\;\pmod{11}$$

Of the above, the number of elements elements that are in $\;\left\{\frac{13}2=6\,,\,7\,,\,8\,,\,9\,\,10\right\}\;$ is $\;3\;$ ,so by Gauss's Lemma we get

$$\left(\frac8{11}\right)=(-1)^3=-1$$

and indeed

$$\left(\frac8{11}\right)=\left(\frac2{11}\right)^3=(-1)^3=-1$$

and now you do the other one.

Answer 2

$a)$ $$\left ( \frac{18}{43} \right )=\left ( \frac{2 \cdot 3^2}{43} \right )=\left ( \frac{2}{43} \right )\left ( \frac{3}{43} \right )^2=(-1)^({43^2-1}{8})=(-1)^{(\frac{42}{8} )\cdot (\frac{44}{8})}=[(-1)^6]^{\frac{11}{2}}=(-1)^{3 \cdot 11}=-1$$

$b)$ $$\displaystyle{\left ( \frac{19}{23} \right )=(-1)^{{9}\cdot{11}} \left ( \frac{23}{19} \right )=(-1)\left ( \frac{4}{19} \right )=(-1)\left ( \frac{2}{19} \right )^2=(-1) \cdot (-1)^{\frac{(19-1) \cdot (19+1)}{8}}= (-1) \cdot (-1)=(-1) \cdot (-1)^{45}=1}$$

The Gauss' Lemma is:

$$\text{Let prime } p>2 \text{ and } p \nmid a. \\If \ m \text{ from the numbers } \\ a \cdot 1, a \cdot 2, \dots , a \cdot \frac{p-1}{2} \text{ have a remainder,when they are divided by } p, \text{ that is between } \frac{p+1}{2} \text{ and } p-1, \text{ then } \left ( \frac{a}{p} \right )=(-1)^m $$

Can you apply it now?