I am reading the wiki article about Quadratic reciprocity and I don’t understand how can I tell if some integer $c$ got quadratic root mod $p$? So far I am using brute search to find $y$ such that $x= c\mod p$ $y^2 \equiv x \bmod p$ for some $y \in \{0,1,\ldots,p\}$ How can I use […]

I am looking for a systematic way of deciding if a given number is a square in $\Bbb Z/n\Bbb Z$. E.g. is $89$ a square in $\Bbb Z/n\Bbb Z$ for $n\in \{25,33,49\}$? Brute-forcing it would take too long here. How else can we do this?

I’m currently studying for exams and this has me stuck. A sample question from a past paper states: Use the quadratic reciprocity theorem to determine whether $11$ is a quadratic residue $\mod p$ for primes of the form: (i) $44k+5$ (ii)$44k+7$ etc… Neither my books or my notes have any specific proofs related to this […]

This is a question from Lang’s ANT, Thm 6, ch.IV, $\S2$. It states that every quadratic extension of $\mathbb{Q}$ is contained in a cyclotomic extension and that it’s a direct consequence of the following result: Let $\zeta_n$ be a primitive $n$-th root of unity for $n$ odd and $$S=\sum_\nu\left(\frac{\nu}{n}\right)\zeta^\nu_n$$ the sum being taken over non-zero […]

The general theorem is: for all odd, distinct primes $p, q$, the following holds: $$\left( \frac{p}{q} \right) \left( \frac{q}{p} \right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}$$ I’ve discovered the following proof for the case $q=3$: Consider the Möbius transformation $f(x) = \frac{1}{1-x}$, defined on $F_{p} \cup {\infty}$. It is a bijection of order 3: $f^{(3)} = Id$. Now we’ll […]

The question is basically the title itself. It is easy to prove using quadratic reciprocity that non squares are non residues for some prime $p$. I would like to make use of this fact in a proof of quadratic reciprocity though and would like a proof that avoids quadratic reciprocity if possible.

Here is the question: Suppose that $p$ is an odd prime. The law of quadratic reciprocity says that $x^2\equiv 2\pmod p$ has a solution. if $p\equiv1 \text{ or } 7 \pmod 8$. Prove that $2^{4n+3}\equiv1 \pmod{8n+1}$. $8n+1$ is a prime I honestly don’t know where to start, I tried starting it with Fermat’s Theorem because […]

Let $f = ax^2 + bxy + cy^2$ be an integral binary quadratic form. We say $D = b^2 – 4ac$ is the discriminant of $f$. If $D < 0$ and $a > 0$, we say $f$ is positive definite. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ […]

If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$. Note that if $p\nmid n$ and $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, then $\left(\frac{n}{p}\right)=\left(\frac{p_1}{p}\right)^{\alpha_1}\left(\frac{p_2}{p}\right)^{\alpha_2}\cdots \left(\frac{p_k}{p}\right)^{\alpha_k}$ $=\left(\frac{p_1}{p}\right)^{\alpha_1\pmod {2}}\left(\frac{p_2}{p}\right)^{\alpha_2\pmod{2}}\cdots \left(\frac{p_k}{p}\right)^{\alpha_k\pmod{2}}$ So i.e. prove exist infinitely many primes $p$ such that $\left(\frac{q_1}{p}\right)\left(\frac{q_2}{p}\right)\cdots \left(\frac{q_n}{p}\right)=-1$, no matter what primes $q_i$ you take.

The definition and properties of Jacobi symbol are stated in this article. I don’t have a textbook handy containing the proofs of the following properties of Jacobi symbol. It seems to me that not many textbooks on elementary number theory contain them. So I think it is nice not only for me but also for […]

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