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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.

Cubic reciprocity is a collection of theorems in number theory that state conditions under which the congruence x³ ≡ p (mod q) is solvable; the word reciprocity comes from the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers and each coprime to 3, x³ ≡ p (mod q) is solvable if and only if x³ ≡ q (mod p) is solvable.

28 questions
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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 sure I completely understand factoring in these…
Dave huff
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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 $q \equiv 2 \pmod 3.$ For primes larger than $3$ with $p \equiv…
Will Jagy
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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 prime, then $2$ is a cubic residue $\pmod p$,…
J. Linne
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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 that I can't get to work. I suspect that the…
Evan Chen
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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 $\chi_{\gamma}(\rho)=\chi_{\rho}(\gamma)$, where…
Srinivasa Granujan
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A proof using cubic reciprocity.

I'm trying to solve this problem: $$n!+n(n+1)/2=m^3.$$ If $n+1=p$ is prime this leads to $2(n-1)!+p=a^3.$ If $p$ is a cubic non-residue for some $q13$. I know we only need to consider…
Fernando Cagarrinho
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Applications of Langlands for GLn Explicit reciprocity laws other than elliptic curves

Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$ for $n \ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves? I've never found such things in articles such like "What is a…
Cloudifold
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When is $2023$ a cubic residue residue mod $p$?

When is $2023$ a cubic residue residue mod $p$ that is $1\pmod{3}$? I do know about quadratic reciprocity, since $x^{2}\equiv2023\pmod{p}$ is only solvable if $p\equiv\pm1\pm3\pm9\pmod{28}$, but I don’t know when $x^{3}\equiv2023\pmod{p}$ is…
Thirdy Yabata
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Is there any way to predict the largest number of consecutive quadratic or cubic residues modulo prime $p$?

We all know that $a$ is a quadratic residue modulo $p$ if and only if $a^{(p-1)/2} \equiv 1 \pmod p$, also $a$ is a cubic residue modulo $p$ if and only if $a^{(p-1)/3} \equiv 1 \pmod p$. Now, for a given prime $p$: (1) is there any way to predict…
عبد الرحمن رمزي محمود
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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 all values of n. I'm trying to exhaust the…
Uday Khanna
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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 $$J(\psi_\pi, \psi_\pi)=-\sum_{i=1}^{p-1}\psi_\pi(i)\psi_\pi(1-i)=\pi?$$ In Lemmermayer's…
Meow
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Cubic non-residue calculation

I am currently studying Cubic residue characters from Kenneth Ireland and Michael Rosen's "A Classical Introduction to Modern Number Theory", and this is the definition given in the book: If $N(\pi) \neq 3$, the cubic residue character of $\alpha$…
Disha
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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 know that applications of quadratic residues are in…
MNic
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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 $\mathbb{F}_{p^2}$ when $p\equiv 13$ or $19\mod24$. Some thoughts: I…
Bob Jones
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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 prime. To accelerate the search, I want to use the…
Peter
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