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2
votes
1answer
35 views

3 is Square in finite field

Let $K$ be a finite field with $p^2$ elements. Show that $3$ is square in $K$. I know that 3 is sum of two squares. Thanks.
3
votes
0answers
61 views

Number Theory: Legendre Symbols

I have the following question. Calculate the Legendre symbol $\bigl(\frac{77}{5^{200}+1}\bigr)$. I know the following: $5^6\equiv1\pmod7$, $5^{10}\equiv1\pmod{11}$. Thus I approached this ...
0
votes
2answers
55 views

Ramification and Quadratic Reciprocity Law

I have a question regarding the follow problem: Show that the prime number 27644437 splits completely in $L = \mathbb{Q}(\sqrt{55})$. From what I understand. This deals with ...
5
votes
0answers
43 views

If $a$ is a quadratic residue modulo every prime $p$, it is a square - without using quadratic reciprocity.

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 ...
0
votes
1answer
23 views

Suplementary law and reciprocity

Let $p$ odd prime. If $\omega\in\overline{\mathbb{F_p}}$ is a 8-th primitive root of unity. Then if $\gamma=\omega+\omega^{-1}$. Why is i) $\gamma^p=\omega^ ...
0
votes
2answers
126 views

Reciprocal of 81 being the sequence of all natural numbers?

According to this document: http://www.answering-christianity.com/fakir60/81.htm describing the theory of scientist Peter Plichta, the reciprocal of 81 is: the ...
-1
votes
1answer
69 views

A Conjecture on The Generalization of Quadratic Reciprocity Law

Is there any way to prove the following conjecture regarding the Generalization of Quadratic Reciprocity Law. The statement being, $$ \left(\dfrac{a_1}{a_2}\right)\left(\dfrac{a_2}{a_3}\right) ...
1
vote
1answer
23 views

Exercise of Quadratic Reciprocity

This is an exercise in Burton : Prove that $$(5/p) =1\ iff\ p\equiv 1,\ 9,\ 11,\ or\ 19\ (20) $$ Note that $5=4+1$ so that $(5/p)=(p/5)$. In further $$ (p/5)^2=(5/p)(p/5) = (-1)^{1\cdot ...
0
votes
0answers
31 views

Find a criterion for the primes p such that (5/p) = 1. [duplicate]

Find a criterion for the primes p such that (5/p) = 1. I don't understand this question it is like Determine all primes P such that (5/p)=1 I appreciate any help
3
votes
3answers
127 views

Elementary, direct proof of when $5$ is a quadratic residue mod $p$

$\newcommand{\kron}[2]{\left( \frac{#1}{#2} \right)}$ It's easy to use Quadratic Reciprocity to show that $\kron{5}{p} = \kron{p}{5} = 1$ when $p \equiv \pm 1 \pmod 5$, and is $-1$ when $p \equiv \pm ...
0
votes
0answers
26 views

Quadratic reciprocity for odd integers

I have a book that proves the three laws of quadratic reciprocity for primes i.e $\left(\frac{-1}p\right)=1$ if $p=1\mod4$ and $-1$ if $p=3\mod4$, where $p$ is prime, etc. But in the end the book ...
0
votes
1answer
39 views

finding all primes $p$ for which a given number is a quadratic residue

I have seen an exercise on the Apostol, but I haven't understood some passages. I would be very grateful if you could solve my doubts. The problem is Find all primes $p$ for which 3 is a ...
0
votes
0answers
20 views

For which positive odd integers $n,$ $(n,33)=1$ does the Jacobi symbol $\left(\frac{33}{n}\right) = 1$?

Attention: What modulus should we use to insure that n is odd? So I know that $\left(\frac{33}{n}\right) = \left(\frac{11}{n}\right)\left(\frac{3}{n}\right)$ then using the legendre properties ...
1
vote
2answers
73 views

factor 9997 using quadratic reciprocity

I looked up the factorization and it is $13\cdot769$, but I have no idea how quadratic reciprocity allows you to deduce this without knowing it. I thought maybe ...
6
votes
1answer
342 views

Is every integer a quadratic residue mod some p?

Is every integer (say $d$) a quadratic residue mod some prime number $p$?
1
vote
0answers
37 views

When can a congruence relation be transformed into quadratic reciprocity expressions?

When can a congruence relation $$p \equiv c_1, c_2, \ldots, c_r \mod{N}$$ be transformed back into quadratic reciprocity expressions $$\left (\frac{d_1}{p} \right) = \left (\frac{d_1}{p} \right) = ...
0
votes
2answers
71 views

Let p be an odd prime, q the smallest quadratic non residue (mod p). Prove q is prime.

So I have this problem; Let p be an odd prime and let q be the smallest positive integer which is a quadratic non residue (mod p). Prove q is a prime. So what I know is that, since q is the ...
2
votes
1answer
157 views

Quadratic reciprocity problem

How can I use quadratic reciprocity to prove that $-3$ is a quadratic residue $\pmod p$ if and only if $p=2$ or $p \equiv 1 \pmod 6$ and deduce that $\mathbb{Z}[\sqrt{-3}]/(p)\cong \mathbb{F}_p ...
0
votes
1answer
45 views

Quadratic congruences problem $(x^2-a)(x^2-b)(x^2-ab)$ ≡ 0 (mod p)

Let p be an odd prime and let a,b ∈ Z such that p ∤ a. Prove that the congruence $(x^2-a)(x^2-b)(x^2-ab)$ ≡ 0 (mod p) is always solvable. Not sure where to begin here.
3
votes
2answers
205 views

elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
1
vote
1answer
69 views

Analytic proof of quadratic reciprocity

Is there any proof of quadratic reciprocity that is more analytic than those described on Wikipedia (http://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity)?
3
votes
1answer
108 views

Consecutive quadratic residues

I was studying quadratic reciprocity laws and came across the following question: Is it true that for every $k \in \mathbb{N} $ there exists a prime $p$ such that $1,2,...,k$ are all quadratic ...
1
vote
1answer
46 views

Quadratic Reciprocity as a consequence of Eisenstein Reciprocity

I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how ...
2
votes
1answer
83 views

Prove that S consists of the quadratic residues mod (p) and T consists of the quadratic non-residues mod (p)

All the followings are $\bmod$ $p$ Let p be an odd prime. Suppose that the set $X = \{ 1,2, . . . , p-1\}$ can be written as the union of two nonempty subsets $S$ and $T$, where $S \neq T$, such that ...
2
votes
1answer
190 views

Quadratic Reciprocity Joke

I found a joke on a site made by user Zev Chonoles on quadratic reciprocity, the joke is as follows: $$\text{Quadratic reciprocity: } \left(\frac{p}{q}\right)=\left(\frac{q}{p}\right), \text{ up to ...
2
votes
1answer
35 views

Jacobi symbol $\left(\frac{(n+1)/2}{n}\right)$

Let $(\frac{a}{n})_J$ be Jacobi symbol defined by \begin{equation} \left(\frac{a}{n}\right)_J=\left(\frac{a}{p_1}\right)^{e_1}\left(\frac{a}{p_2}\right)^{e_2}\cdots\left(\frac{a}{p_k}\right)^{e_k} ...
1
vote
2answers
40 views

Number of elements that exist so $b^3 \equiv a\pmod n$, when n is composed of p and q who are prime numbers

Given $2$ prime numbers,$ p$ and $q$, that are both not even, and $3$ doesn't divide $p-1$ or $q-1$, and $n=pq$, how many elements in $Z^*_n$ exists that has $b$ such that $b^3\equiv a\pmod n$ . I ...
5
votes
6answers
368 views

What is the point of quadratic residues?

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most ...
0
votes
1answer
110 views

Legendre symbols (p/q) = (a/q) [duplicate]

Suppose q and p are odd primes and p = q+4a for some integer a. From the properties of the Legendre symbol and the Law of Quadratic Reciprocity prove that the following two identities hold. (p/q) = ...
0
votes
2answers
53 views

$2^{4n+1} \equiv 1 \pmod{8n+7}$, this has been bugging me

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 ...
0
votes
2answers
78 views

If $x^2 \equiv a \bmod p$ has a solution, does it necessarily have infinitely many solutions? [closed]

If not, how would you prove it for a specific case? Such as, for example, $x^2 \equiv 2 \bmod 7$ ?
1
vote
1answer
58 views

Suppose $p$ and $q$ are odd primes and $p = q + 4a$ for some $a$. Prove that $(\frac ap) = (\frac aq)$ holds. [duplicate]

This problem also had me prove that $(\frac pq) = (\frac aq)$, but I've already managed to do that. I've tried messing around with the Law of Quadratic Reciprocity but can't get anything. I've also ...
0
votes
2answers
29 views

Quadratic character of 3

Using the QRL prove that, for any odd prime $p$, $(3/p) = 1$ if $p$ is congruent to $1$ or $11 \pmod{12}$. Using the Quadratic reciprocity law, $(3/p)(p/3)=(-1)^{(3-1)(p-1)/4}$, I get that the ...
1
vote
1answer
39 views

Prove that $\left(\frac2p\right) = 1$ if $p \equiv 1,7 \pmod 8$ and $\left(\frac2p\right) = -1$ if $p \equiv 3,5 \pmod 8$ using ring theory

Let $p$ be an odd prime number and let $\alpha = [X] \in R=\mathbb F_p[X]/\langle X^4+1\rangle$, and $y = \alpha + \alpha^{-1}$ I've proven: 1) $\alpha$ is a primitive eight root of unity in $R$. ...
0
votes
0answers
164 views

Quadratic Reciprocity [duplicate]

For any integer $a$ and any two primes $p,q$ with $(p,q)=1$. Prove that if $p \equiv q\pmod {4a}$, then $\displaystyle\left(\frac ap\right)=\left(\frac aq\right)$ I know I need to write $a=(−1)^{e_0} ...
1
vote
1answer
146 views

Quadratic Reciprocity

For any integer $a$ and any two primes $p,q$ with $(p,q)=1$. Prove that if $p \equiv q$ mod $4a$, then $(\frac{a}{p})=(\frac{a}{q})$ I know I need to write $a=(−1)^{e_0} 2^{e_2} p_1 p_2 \cdots p_r$ ...
1
vote
0answers
50 views

Show that $x^2\equiv a \pmod {2^n}$ has a solution where $a\equiv1 \pmod 8$ and $n\ge3$

Show that $$x^2\equiv a \pmod {2^n}$$ has a solution where $a\equiv1 \pmod 8$ and $n\ge3$ Actually, the question I had to solve was more complicated something like this: $x^2\equiv a ...
4
votes
2answers
148 views

Show that prime $p=4n+1$ is a divisor of $n^{n}-1$

Show that the prime number $p=4n+1$ is a divisor of $n^{n}-1$ Ok, the question itself is simple as hell, but I couldn't think of a simple way to solve this question. I tried to solve the ...
2
votes
0answers
161 views

When quadratic formula $ax^2 + bx +c = 0$ (mod p) is solvable modulo prime p?

I am learning about quadratic congruences and I don't now how to decide, for which a,b,c and p there is a solution of the congruence. It is sufficient if the determinant $\sqrt{b^2-4ac}$ has a ...
0
votes
0answers
36 views

How do I prove that $x^2\equiv-1\pmod{p}$ [duplicate]

How do I prove that $x^2\equiv-1\pmod{p}$ iff $p$ is prime at form: $p=4n+1$. I have to use Wilson theroem... (I'm asking this, becuase I didn't understand it from my prev. Q, how it proves that ...
0
votes
1answer
507 views

Solve quadratic congruence using the Chinese Remainder Theorem

Solve $x^2 \equiv$ -1 (mod 205) I have that as 205 = 5 x 41 this becomes: $x^2$ $\equiv $ -1 (mod 5) $x^2$ $\equiv $ -1 (mod 41) then i'm not sure where to go. I can see 32 is a solution but I ...
2
votes
2answers
132 views

Number Theory: Solutions of $ax^2+by^2\equiv1 \pmod p$

Assume $p$ is a prime number and $\gcd(ab, p)=1$. Show that the number of integer solutions $(x, y)$ of $ax^2+by^2 \equiv 1 \pmod p$ is $$p - \left(\dfrac{-ab}{p}\right)$$ where ...
0
votes
1answer
53 views

Proving something with Wilson's Theorem [continued.]

At first I asked this: Proving something with Wilson theorem. Now I have to prove that if $p=4n+3$ it's impossible to represent $-1$ in the form $x^2$ modulo $p$. How can I prove it? Thank you!
2
votes
2answers
122 views

Proving something with Wilson theorem

I need to prove that $x^2\equiv -1\pmod p$ if $p=4n+1$. ($p$ is prime of course...) I need to use Wilson theorem.
2
votes
1answer
172 views

A condition for an odd prime to be represented by a binary quadratic form of a given discriminant

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 ...
0
votes
1answer
846 views

Proofs of the properties of Jacobi symbol

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 ...
1
vote
1answer
130 views

Quadratic Reciprocity - Legendre Symbols

Find the value of $((1\cdot 2)/73)+((2\cdot 3)/73)+...+((71\cdot 72)/73)$. This is based off each fraction being a Legendre Symbol. I tried to find a pattern... but I could't find anything. Also, I ...
5
votes
0answers
90 views

What should be taught about the reciprocity law for high school gifted student

This autumn I have to teach a mini course for a small group of high school student (mathematical gifted class) on Quadratic Reciprocity Law (3 lectures in ten hours). This is the requirement of the ...
8
votes
1answer
175 views

Quadratic extensions of $\mathbb Q$

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 ...
2
votes
1answer
67 views

Find $m, n$ such that $\frac{n^2 + 1}{m^2 + 1 }$ is an integer multiple of a perfect square

I'm trying to find $n,m \in \mathbb{N}$ such that $\sqrt{ \frac{n^2+1}{2(m^2+1)}}$ is rational. I see that if $a,b$ are relatively prime $\sqrt{ \frac{a}{b}}$ is rational if and only if $a,b$ are ...