Questions tagged [cubic-reciprocity]

Question 1

In Diamond and Shurman, I am trying to understand the weak form of cubic reciprocity in section 4.11 - how it comes from the more standard version. In the following $A$ is the Eisenstein integers, and ...

Question 2

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 $q<p$ the equation has no solution. This seems to ...

Question 3

Proposition: If $p \equiv 1 \pmod 3$, then $x^3 \equiv 2 \pmod p$ is solvable if and only if there are integers $C$ and $D$ such that $p=C^2 + 27 D^2$. Proof: If $x^3 \equiv 2 \pmod p$ is solvable, ...

Question 4

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 ...

Question 5

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 ...

Question 6

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’...

Question 7

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 ...

Question 8

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 ...

Question 9

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. $\...

Question 10

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 ...

Question 11

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)...

Question 12

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 ...

Question 13

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 ...

Question 14

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 ...

Question 15

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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Questions tagged [cubic-reciprocity]

Weak cubic reciprocity in Diamond and Shurman

A proof using cubic reciprocity.

Cubic Character of 2

Applications of Langlands for GLn Explicit reciprocity laws other than elliptic curves

Cubic non-residue calculation

When is $2023$ a cubic residue residue mod $p$?

Is there any way to predict the largest number of consecutive quadratic or cubic residues modulo prime $p$?

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 $6k+ 1$?

If $x^3+2y^3=p$ has a solution over positive integers, then is it unique?

Perfect squares and cubes in quadratic number fields

Extending the Supplement of Eisenstein Reciprocity

Reference Request: Cubic and Biquadratic Reciprocity Law

Are cube roots evenly distributed modulo primes?

Applications of higher order reciprocity laws

Quadratic and Cubic Fractional Residues

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