Questions tagged [algebraic-number-theory]
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
2,171
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3
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Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?
The solvable Emma Lehmer quintic is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + ...
- 12.1k
0
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0
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148
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examples of class fields
Can anyone explain with a numerical example of generating class field with Kummer extension? I have not come across any standard reference which does give examples. Please help or cite any reference ...
2
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0
answers
132
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$\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$
I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?
- 21
8
votes
2
answers
650
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Adjoining torsion points from abelian varieties
Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically ...
- 11.2k
6
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1
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781
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How to test if the power of some algebraic number is the rational combination of two specific algebraic numbers?
Suppose we are given three algebraic numbers $\alpha,\beta,\gamma$ by presenting their minimal polynomial (degree less than $m$), the goal is to compute all positive integers $n$ such that $\alpha^n$ ...
- 1,493
48
votes
4
answers
4k
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Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.2k
6
votes
2
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373
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On a minimal algebraic number field which satisfies the principal ideal theorem
By an algebraic number field, we mean a finite extension field of the field of rational numbers.
Let $k$ be an algebraic number field, we denote by $\mathcal{O}_k$ the ring of algebraic integers in $k$...
- 1,159
2
votes
1
answer
250
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not Gauss sum with the same magnitude
Gauss sum is a sum of $p$ roots of unity with magnitude $\sqrt{p}$. Does another sum with such property exist?
More exactly. Let $p$ be a prime number. $\zeta^p=1,\;\zeta\ne 1$. Causs sum: $G=\sum_{...
- 81
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3
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References for $K_{4k}(\mathbb{Z})$
Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
- 17.5k
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284
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The number of solutions of a Diophantine equation [closed]
Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...
- 11.2k
4
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0
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327
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Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.2k
5
votes
1
answer
515
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Disjoint images of polynomials
Are there any $f,g \in \mathbb{Q}[x]$ such that for every root of unity $\zeta$, and every $a,b \in \mathbb{Q}(\zeta)$, $f(a) \neq g(b)?$
- 11.2k
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Is a number field uniquely determined by the primes which split in it?
Let $K/\mathbb{Q}$ be a number field. We say that a rational prime $p$ splits in $K$ if there exists a prime $\mathfrak{p}$ of $K$ above $p$ of interia degree $1$.
Is a number field $K$ ...
- 20.8k
1
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164
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Slope decomposition of a product of operators
I'm trying to relate the slope decomposition of a product of linear operators to the slope decompositions with regard to each of the operators in the product.
First I'll give some background, for ...
- 787
1
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0
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150
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Skew symmetry for the Hilbert symbol
Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ $$(a,...
- 33
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3
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Textbook request for class field theory [duplicate]
I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...
2
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0
answers
179
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What are the minimal degrees of the real and imaginary part of an algebraic complex number? [closed]
Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...
- 13.2k
19
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1
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A good book on adeles and ideles
Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...
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3
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1
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282
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What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?
I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of $A$-modules, where $A$...
- 169
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votes
1
answer
760
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Intersection of a ring class field of a quadratic field K with the cyclotomic extension of K
Let $K$ be a quadratic field. Let $f\in\mathbb{Z}_{\geq 1}$. Let $\mathcal{O}_f=\mathbf{Z}+f\mathcal{O}_K$ be the unique order of $K$ of index $f$ in $\mathcal{O}_K$. Let $H_f^{ring}$ denote the ring ...
- 7,521
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2
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What is an infinite prime in algebraic topology?
The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...
- 4,730
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0
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398
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Rings of algebraic integers as quotients of polynomial rings
The ring of integers $\mathcal{O}_K$ of a number field $K$ is always isomorphic to some ring of the form $\mathbb{Z}[x_1, ..., x_r]/\mathfrak{p}$, where $\mathfrak{p} \subset \mathbb{Z}[x_1, ..., x_r]$...
- 704
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0
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131
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Reciprocity laws in different dimensions
Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to −\...
- 33
1
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0
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100
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Relation between 1-dimensional and 2-dimensional reciprocity maps
Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, \...
- 33
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votes
0
answers
695
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The construction of the 257gon
If $\zeta\in\mathbb C$ is a primitive $257$th root of unity, the Galois group $\operatorname{Gal}(\mathbb Q(\zeta)/\mathbb Q)$ is cyclic of order $256=2^8$, so we know that there is a sequence of $8$ ...
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6
votes
2
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480
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Iwasawa's mu-invariant for noncyclotomic $\mathbf{Z}_p$ extensions of cyclotomic fields?
Let $p$ be an odd prime number, $m$ a positive integer with $p\mid m$. Put $k=\mathbf{Q}(\mu_m)$.
(1) Is there any example where certain noncyclotomic $\mathbf{Z}_p$-extension $k_\infty/k$ has ...
- 61
2
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0
answers
136
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Lowest degree polynomial with integer coefficients yielding $1/\sqrt{2^n}$
Let $x = \cos(\pi/8) = \frac{1}{2} \sqrt{2+\sqrt{2}}$ and $y = \sin(\pi/8) = \frac{1}{2} \sqrt{2-\sqrt{2}}$. What is the lowest degree polynomial $p(x,y)$ with integer coefficients such that $p(x,y) = ...
- 315
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1
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154
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Units of an extension of $\mathbb{Z}$ [closed]
Let $P(x)\in\mathbb{Z}[x]$ be monic and irreducible over $\mathbb{Q}[x]$, and let $\theta$ be a root of $P(x)$. Let $K = \{a + b\theta\} \subseteq \mathbb{Z}[\theta]$. When is it the case that there ...
- 1,831
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2
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Galois groups and prescribed ramification
What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...
- 11.2k
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2
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817
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How did height in algeb. number theory/elliptic curves started?
Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...
- 2,175
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0
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492
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Motivating mathematics(particularly algebraic number theory) through historical problems [closed]
Most mathematical textbooks start a subject by going backwards, historically. They will define the terms that were invented to solve a problem in their polished form and then use these definitions and ...
5
votes
2
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615
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Why is the supersingular locus the zero locus of a modular form?
This question is related to my other question here: Examples of subspaces singled out by modular forms.
Here I am wondering if there is a philosophical explanation about why the supersingular locus ...
- 231
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4
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Can a sum of roots of unity be an integer?
Let $n \geq 2$, $H \lneq (\mathbb{Z}/n\mathbb{Z})^*$, $\zeta_k$ a primitive $k$-th root of unity. Is it possible that $$\sum_{h \in H} \zeta_k^{h} \in \mathbb{Z}$$ for every $k$ dividing $n$ such that ...
- 11.2k
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votes
2
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334
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explicit uniformizer for the false Tate extension
Let $p$ be an odd prime and let $n\geq 1$. Set $K=\mathbb{Q}_p(\zeta_{p^n})$,
$L=\mathbb{Q}_p(\sqrt[p^n]{p})$, and $M=KL$. I claim that $M$ is totally ramified of degree $\phi(p^n)p^n$ (the proof ...
- 7,521
2
votes
0
answers
57
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Semi-simple controlling operator
I've just come across this paper by Coleman and Edixhoven called "On the semi-simplicity of the $U_p$ operator on modular forms", where (as the title says) they show that the $U_p$ operator is semi-...
- 787
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0
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56
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Image of the typenorm contains the squares
I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...
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139
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Results for resolution of equations in polynomial ring
Is there any reference for resolution of equations in a polynomial ring, such as $x^2+y^2=z^2$ in $\mathbb{C}[t]$? Thanks!
- 75
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304
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Without Skolem–Mahler–Lech Theorem? [closed]
Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$.
Is there a "simple way" to prove this ...
- 41
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2
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Is there an algebraic number that cannot be expressed using only elementary functions?
(this is basically a repost of a question I asked at M.SE last year)
Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that ...
- 6,709
1
vote
0
answers
119
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The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials
Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...
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Can there be a power basis for a totally real field of high degree?
A number field $K$ is said to have a power basis if there is an $\alpha \in K$ such that the full ring of integers $O_K$ is the $\mathbb{Z}$-linear span of $1,\alpha,\alpha^2,\ldots,\alpha^{\deg{K}-1}$...
- 13.7k
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0
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90
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Characterize the set of roots of cubics with certain properties
Let $P(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $3$. Suppose that $\alpha_1, \alpha_2, \alpha_3$ are roots of $P(x)$. For what such $P(x)$ is it the case that the ring of integers ...
- 1,831
5
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0
answers
757
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Is there an excplicit number field of definition for an Abelian Variety $A/\mathbb{C}$ with CM?
Consider a simple abelian variety $A/\mathbb{C}$ with sufficiently many CMs by $\mathcal{O}$, where $\mathcal{O}$ is an order in a CM field $K$. Specifically, $K$ is a CM field of degree $2g$, where $...
- 1,443
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votes
3
answers
2k
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Ideal classes fixed by the Galois group
Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...
- 1,001
14
votes
3
answers
915
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Asymptotics for algebraic numbers of height less than one
The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$?
The rather ...
- 13.7k
7
votes
0
answers
548
views
Factors of the polynomial $X^n-a$
I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...
- 7,652
7
votes
3
answers
827
views
Constructing quintic number fields with certain splitting behaviour
I am looking for number fields $K$ which satisfy the following properties:
$[K:\mathbb{Q}]=5$.
The Galois closure of $K$ has Galois group $S_5$.
For each prime $p$ which ramifies in $K$, there exists ...
- 20.8k
1
vote
1
answer
1k
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A letter from J. P. Serre
Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
- 61
5
votes
2
answers
399
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Time-line until the publicaton of Weil of "Numbers of solutions of equations in finite fields"
In "On the history of the Weil Conjectures" Dieudonné says:
"Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...".
C. F. Gauss, ...
- 61
4
votes
1
answer
591
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How frequently is 3 a cubic residue mod primes in an arithmetic progression?
Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$?
Or, an equivalent formulation using quadratic forms: ...
