Questions tagged [cubic-reciprocity]

Use this tag for questions about theorems in number theory that state conditions under which the congruence x³ ≡ p (mod q) is solvable.

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Are primes of the form $6k+1$ a cube modulo $n$, if $3\nmid n$ and none of the prime factors of $n$ is of the form $6k+ 1$?

I wonder if we can assume the following statement to be true in general: Let $p$ be a prime of the form $6k+1$ and $n<p$ a natural number less than $p$. If $3$ does not divide $n$ and none of the ...
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If $x^3+2y^3=p$ has a solution over positive integers, then is it unique?

I am trying to prove that for an odd prime $p$, if $x^3+2y^3=p$ has a solution over positive integers, then it is unique. My work, Assume $(x,y)=(a,b)$ and $(x,y)=(c,d)$ are different solutions. $\...
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Perfect squares and cubes in quadratic number fields

Suppose we are given a quadratic number field $\mathbb{Q}(\sqrt{d})$, for some integer $d$ which is not a perfect square. I wish to study when is an element $\alpha \in \mathbb{Q}(\sqrt{d})$ a perfect ...
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Extending the Supplement of Eisenstein Reciprocity

One of the supplements of Eisenstein Reciprocity states the following: Supplement: If $m$ is an odd prime and $a$ is a rational integer relatively prime to $m$, then $\left(\frac{1-\zeta_m}{a }\right)...
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Reference Request: Cubic and Biquadratic Reciprocity Law

I want to read about the Cubic and Biquadratic Reciprocity Laws after learning the Quadratic Reciprocity Law. I already know about Franz Lemmermeyer's book "Reciprocity Laws", but I think this is a ...
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Are cube roots evenly distributed modulo primes?

Are the cube roots of two integers chosen from a uniform distribution between $1$ and $p-1$ inclusive, $p$ prime, essentially evenly distributed? Note that I will use a $p$ such that $p$ is not ...
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Applications of higher order reciprocity laws

I'm currently studying quartic & cubic residues and their reciprocity laws, and would like to know of any real world applications to finding the values of their respective residue symbols. I ...
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Quadratic and Cubic Fractional Residues

Is there any method to find out the characterization of all primes $p$ such that $\frac{a}{b}$ is a quadratic residue modulo $p$ such that $a$ and $b$ are primes? Is there any method to do the same ...
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$n$th power residue conjecture

Let $\left(\dfrac{a}{p}\right)_n$ be the $n$th power residue symbol such that when $a$ is an $n$th power residue $\pmod p$, $\left(\dfrac{a}{p}\right)_n=1$ otherwise $\left(\dfrac{a}{p}\right)_n=\zeta$...
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Quintic reciprocity conjecture

Let $p=x^4 + 25x^2 + 125$ be a prime. Prove that $2$ is a quintic residue $\pmod p$, and therefore $y^5=2\pmod p$ is solvable. A similar example was first conjectured by Euler: If $p=x^2 + 27$ is a ...
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For what values of n is $f(x) = x^3 mod(n)$ a bijection from $X={(0,1,2,...,n-1)}$ to itself.

I was thinking about shuffling, mapping ${(1,2,...,n-1)}$ to a permutation of itself using a mapping like $x\to x^k \mod(n)$ and clearly $k=2$ cannot work since $1$ and $n-1$ have the same image for ...
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$\sqrt{2}+1$ is a cube in $\mathbb{F}_{p^2}$ when...

I have a conjecture and I think I have a class field theory proof of it, but I would like to know if there's a QR or CR proof of it. The statement is that $\sqrt{2}+1$ is a cube in $\mathbb{F}_{p^2}$ ...
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Proofs regarding cubic reciprocity

How do I go about these proofs? I can't find much online about cubic reciprocity. Suppose $p \geq 5$. If $p \equiv 1, 7 \pmod{12} $, then the number of distinct nonzero cubic residues (mod p) is $(p-...
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How to factor in cubic extensions?

Working within the field $K=\mathbb{Q}(\sqrt[3]{n})$, for any cube root of $n$, how does one factor the unramified rational prime ideals $(p)$? For starters, I'm relatively new to this and not too ...
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Given an integer x^3 ≡ n (mod p), can we find x?

Okay, I have this for a class homework question. If we are given n and p, I know a few things: If p≡1 (mod 3), then we can use the theorem of cubic reciprocity to determine whether there exists an ...
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Artin Reciprocity $\implies$ Cubic Reciprocity

I'm trying to understand the proof of cubic reciprocity from Artin reciprocity as outlined in this well-known previous math.SE question and the link KCd mentions there. However, there's one final step ...
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Can I restrict the possible factors of $2\uparrow \uparrow 4+3\uparrow \uparrow 4$?

I would like to accelerate the search of prime factors of $$2\uparrow \uparrow 4+3\uparrow \uparrow 4$$ In a question, I asked for a prime factor and another user also asked, whether this number is ...
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2 votes
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Cubic character of the third root of unity

I'm working on an exercise from Ireland and Rosen, and I want to know if I'm on the right track. Let $\omega = \frac{ -1 + \sqrt{-3}}{2}$, and consider $\mathbb{Z}[\omega]$. Let $\gamma$ be a primary ...
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Cubic Jacobi sum

If $\psi_\pi= \left( \frac{-}{\pi} \right)_3$ is the cubic residue character and $\pi$ is a prime with $\pi \not \equiv0 \text{ mod }3$, then is it true that $$J(\psi_\pi, \psi_\pi)=-\sum_{i=1}^{p-1}\...
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2 votes
1 answer
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Cubic residues over $\mathbb{Z}_{p^2}^{*}$

Definition: $x\in\mathbb{Z}_{n}^{*}$ is a cubic residue if there exists $y\in\mathbb{Z}_{n}^{*}$ s.t. $y^3\equiv x \pmod{n}$. I have been asked to prove (and I already did) that if $n=pq$, where $p\...
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Previous step of the supplement to the law of cubic reciprocity

Let $\gamma$, $\rho\in\mathbb{Z[\omega]}$ be different primary irreducibles (i.e. $\gamma$, $ \rho\equiv 2(3)$), where $\omega=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$. I have to prove that $\chi_{\gamma}(\...
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10 votes
1 answer
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Cube roots of five

This is not really homework. I might be able to do this myself in time, from the methods in Ireland and Rosen. Note that every number has exactly one cube root $\pmod q$ for any prime $q \equiv 2 \...
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