2 Quadratic Reciprocity Proofs

**swashbucklord** · June 19th 2011, 02:19 PM

Hi all, messed up using Latex at first. Okay, got two problems on quadratic reciprocity. I've worked up something of a proof for each, but I feel very uncertain. Thought maybe someone could give me some pointers.

Okay, simple one first:
Suppose p is a prime such that $p\equiv 3 (mod 4)$ , and let a be a QR modulo p.
Show that $x=a^\frac{p+1}{4}$ is a solution to
$x^2\equiv a(mod\, p)$

My proof:
a - QR mod p $\Rightarrow$ the Legendre symbol
$(\frac{a}{p})= 1 \equiv a^{\frac{p-1}{2}}$ ,
so clearly $1\equiv a^{\frac{p-1}{2}} (mod\, p)$ .
Multiplying both sides by a yields
$a\equiv a^{\frac{p-1}{2}}*a\equiv a^{\frac{p+1}{2}}$ .
But if $x=a^{\frac{p+1}{4}}$ ,
we have that $x^{2}=a^{\frac{p+1}{2}}$ .
We know that $a^{\frac{p+1}{2}}$ is congruent to a modulo p, so
$x=a^{\frac{p+1}{4}}$ is a solution to the congruence.

My trouble is that this seems too easy, and not once did I apply $p\equiv 3(mod\, 4)$ .

Any pointers? Thanks in advance.

**chisigma** · June 19th 2011, 04:56 PM

The 'solution' $x= a^{\frac{p+1}{4}}$ must of course be an integer and that happens only if $p \equiv 3\ (\text{mod}\ 4)$ ...

Kind regards

$\chi$ $\sigma$

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