# A proof of a property of the Legendre symbol.

My book mentioned the following property of Legendre symbol:

$$\left(\frac{a^2}{p}\right) =1,$$

And it said in the proof That the integer a trivially satisfies the congruence $$x^2 \equiv a^2 \pmod{p}$$; hence, $$\left(\frac{a^2}{p}\right) =1.$$

But I do not understand how "a trivially satisfies the congruence $$x^2 \equiv a^2 \pmod{p}$$", could anyone explain this for me please?

• All that says is that $a^2\equiv a^2\pmod p$ – lulu Jun 12 at 10:15

$$\left(\dfrac{k}{p}\right)$$
is defined to be $$1$$ if there exists $$x$$ such that $$x^2\equiv k\mod p$$. When $$k=a^2$$, we can choose $$x=a$$ and thus $$x^2\equiv a^2\equiv k\mod p$$; thus by definition the Legendre symbol is 1.
Another argument is, that we have $$\left(\frac{a^2}{p}\right) =\left(\frac{a}{p}\right)^2=(\pm 1)^2=1$$ with $$p\nmid a$$, because the Legendre symbol is multiplicative.