Questions tagged [quadratic-reciprocity]

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. (Ref: http://en.m.wikipedia.org/wiki/Quadratic_reciprocity)

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Reference: Wikipedia

The theorem was conjectured by Euler and Legendre and first proven by Gauss. He refers to it as the "fundamental theorem" in the "Disquisitiones Arithmeticae".

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Uses of quadratic reciprocity theorem

I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic enough to be shown immediately when presenting the…
Gadi A
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How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow GL_1(\mathbb{C})$ be a $1$-dimensional representation…
Nicole
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Explicit formula for Fermat's 4k+1 theorem

Let $p$ be a prime number of the form $4k+1$. Fermat's theorem asserts that $p$ is a sum of two squares, $p=x^2+y^2$. There are different proofs of this statement (descent, Gaussian integers,...). And recently I've learned there is the following…
Grigory M
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A particular case of the quadratic reciprocity law

To motivate my question, recall the following well-known fact: Suppose that $p\equiv 1\pmod 4$ is a prime number. Then the equation $x^2\equiv -1\pmod p$ has a solution. One can show this as follows: Consider the following polynomial in ${\mathbb…
Bruce George
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Zolotarev's Lemma and Quadratic Reciprocity

The law of quadratic reciprocity is unquestionably one of the most famous results of mathematics. Carl Gauss, often called the "Prince of Mathematicians", referred to it as "The Golden Theorem". He published six proofs of it in his lifetime. To date…
David Reed
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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 quadratic reciprocity though and would like a proof…
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Special Cases of Quadratic Reciprocity and Counting Fixed Points

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…
Ofir
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Is there a proof of quadratic reciprocity using $p$-adic numbers?

I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can get it by considering the cyclotomic extension of $\mathbb{Q}_{p}$. However, I want to know if there's any proof of…
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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 important in number theory?
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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 following result: Let $\zeta_n$ be a primitive $n$-th…
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Primes of the form $p=a^2-2b^2$.

I've stumbled upon this and I was wondering if anyone here could come up with a simple proof: Let $p$ be a prime such that $p\equiv 1 \bmod 8$, and let $a,b\geq 1$ such that $$p=a^2-2b^2.$$ Question: Is $b$ necessarily a square modulo $p$? I have…
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Is every integer a quadratic residue mod some p?

Is every integer (say $d$) a quadratic residue mod some prime number $p$?
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If $d$ divides $a^4+a^3+2a^2-4a+3$, prove that $d$ is a fourth power modulo $13$

If $d$ divides $f(a)=a^4+a^3+2a^2-4a+3$, prove that $d$ is a fourth power modulo $13$. $f(a)\equiv{(a-3)}^4\pmod {13}$. But how can we prove any divisor of $f(a)$ is a fourth power? If we prove that any prime divisor $p$ of $f(a)$ is a fourth power…
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Irreducibility in a polynomial related to quadratic residues

From Romania TST 2004 Day 5 P3, I was introduced to the polynomial $$f(x)=\sum_{i=1}^{p-1} \left( \frac{i}{p} \right)x^{i-1}$$ This polynomial is clearly not irreducible - $x=1$ is a root. Even more - this MO thread gives that for $p \equiv 1…
Gyumin Roh
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How can one deduce quadratic reciprocity from Hilbert reciprocity?

Hilbert reciprocity says the following: Define $(a,b)_p$ to be $1$ if there is a non-trivial solution in $\mathbb{Q}_p$ to $z^2=ax^2+by^2$, and $-1$ if there isn't. Then $\prod_p (a,b)_p =1$, where the product runs also over the infinite prime (and…
Nicole
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