All Questions
Tagged with nt.number-theory algebraic-number-theory
1,422 questions
1
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The value of the Hauptmodul at CM point
Let $J$ be a classical normalized $j$-invariant (that is, J=j-744).
Then, it is a classical result that $J(\tau)$ is an algebraic integer if $\tau$ is an imaginary quadratic number (that is, $a\tau^2+...
- 111
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0
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99
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On the form of algebraic numbers belonging to a specific field extension
Let $m>1$ be an integer and set $\theta=10^{-1/m}$. For a $\gamma\in \mathbb{Q}(\theta)$, there exists $a_0,\ldots,a_{m-1}\in \mathbb{Q}$ such that
$$
\gamma=a_0+a_1\theta+\cdots+a_{m-1}\theta^{m-1}...
- 515
2
votes
1
answer
126
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Changing the weight space for an eigenvariety
Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
- 93
3
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2
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360
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Largest prime factors of integer polynomials
I have a question in analytic number theory which is closely related to the open problem (Bunyakovsky conjecture and more generally, Schinzel's hypothesis H) that asks you if, any irreducible ...
- 302
3
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1
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121
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Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
1
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1
answer
240
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The equation $ax^2 +by^2 =1 \mod P$ in cyclotomic field
Let $L$ be a cyclotomic field, and $P$ a prime ideal of $\mathcal{O}_L$.
is there any symbol for the equation $ax^2 + by^2 =1 \mod P$ and if so, is it computable in polynomial time?
if $a$ is ...
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175
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Modularity from cubic reciprocity: does it generalize?
Background. Let $a$ be a cubefree integer, $N=3\prod_{p\mid a}p$ and
$$
a_p=\#\{\text{solutions to}\,\, x^3\equiv a\,\,(\text{mod}\,\, p)\}-1.
$$ Let $\zeta$ be a primitive cube root of unity and $A=\...
- 171
1
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2
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268
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Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that
The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
- 505
8
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1
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868
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Brauer–Siegel's Theorem and application
$\newcommand\underto[1]{\xrightarrow[#1]{}}$Brauer–Siegel's Theorem says that if we consider an infinite sequence of number fields $K_i/\mathbb{Q}$ such that $$\frac{[K_i:\mathbb{Q}]}{\log{|D_i|}}\...
- 103
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89
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Some invariant subsets of roots of unity
Let $n$ and $t$ be positive integers with $\gcd(t,n)=1$, let $C$ be the collection of all primitive $n$th roots of unity and let $S$ be a subset of $C$ such that, for every $\ell\in\mathbb{N}$,
$$(1)\...
- 934
3
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0
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150
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$p$-adic points of open subschemes of complete intersections
I'm currently studying the $3\times 3$ magic square of squares problem for a research project. The variety is initially given by the intersection of $8$ quadrics in $\mathbb{P}^8$, but via Gröbner ...
- 31
5
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2
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343
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Non-commuting elements of finite orders in a reductive group over a p-adic field
Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma:
Lemma. Assuming that $p$ is "good" for $G$,...
- 14.1k
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1
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112
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Ray class field and its conductor
Let $\mathfrak{m}$ be an ideal of the ring of integers of a number field $K$ and $S$ is a subset of the real places, then the ray class group of
$\mathfrak{m}$ and $S$ is the quotient group
$$I^{\...
- 287
3
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222
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Linear independence of $(n^\alpha)$ when $\alpha$ is an irrational number
Besicovitch proved in 1940 in the paper 'On the linear independence of fractional powers of integers' https://doi.org/10.1112/jlms/s1-15.1.3 that if $\alpha$ is a non-integer rational number then $S_\...
- 184
3
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1
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203
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Chowla's theorem on class number of real quadratic field
Let $p\equiv1\bmod 4$ be a prime number and $h$
the class number of real quadratic field $\mathbb Q(\sqrt{p})$, $\epsilon=\frac{t+u\sqrt{p}}{2}$ its fundamental unit. In this paper https://www.pnas....
- 287
2
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1
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249
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A question on distinguished pairs
I am reading Alexandru, Popescu, and Zaharescu, "On the Closed Subfields of $\mathbb{C}_p$" (see https://tinyurl.com/kknmzbyx). The authors give the following definition:
Let $\alpha, \beta \...
anon
11
votes
1
answer
598
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How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
- 113
3
votes
3
answers
382
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On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
1
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0
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90
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Can computer algebra system compute Galois group/splitting field of a polynomial over $p$-adic number field of higher degree?
I am looking for a computer algebra system that checks if the splitting fields of two polynomials over a $p$-number field are the same or not. At least, knowing their splitting fields are isomorphic ...
- 195
2
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1
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157
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$f(x)\bmod p$ and decomposition of prime ideals
While reading Serre's beautiful book Lectures on $N_X(p)$, I thought of a related question.
Let $f(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial with integer coefficients. Let $K$ be the ...
- 709
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1
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159
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Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
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1
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150
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On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
- 17
1
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0
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49
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Subgroups of $K^n$ and $GL_n(\mathbb{A_K})$
Throughout the question, $v$ is an index for the finite places of a number field $K$, and $\mathfrak{p}_v$ denotes the associated prime ideal. It is a classical fact that there is a map from $\mathbb{...
- 253
3
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0
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71
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About the index of an inverse limit related to the tame inertia group
Let me begin with the context: $p$ is a prime number. We consider the extensions
$$ \mathbb{Q}_p\subset \mathbb{Q}_p^{nr}\subset \mathbb{Q}_p^{tm}\subset \overline{\mathbb{Q}}_p,$$
where $\mathbb{Q}_p^...
3
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1
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178
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Galois cohomology of imaginary abelian number fields
Let $K$ be an imaginary abelian number field of degree $[K:\mathbb{Q}] \geq 4$. Let $G=\text{Gal}(K/\mathbb{Q})$, and denote by $U_K$ the unit group of $K$. Can we show that the order of the first ...
- 437
2
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1
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113
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Inequality between archimedean and non-archimedean height function on number fields
Let $K$ be a number field, $V=V(K)$ be the set of valuations of $K$,
$V_0$ be the set of non-archimedean valuations of $K$,
$V_1$ be the set of archimedean valuations of $K$.
For any $x\in K^\times$, ...
- 95
2
votes
1
answer
175
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Relation between the genus number and the ambiguous class number
It is well known that for $K/F$ a finite "cyclic" extension of number fields, we have $g_{K/F}=a_{K/F}$, where $g_{K/F}$ denotes the relative genus number, and $a_{K/F}$ denotes the ...
- 437
2
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1
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101
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Bound on number of extensions of Q unramified outside a fixed prime
Are there any known asymptotic bounds on the number of degree $d$ extensions of $\mathbb{Q}$ unramified outside a fixed prime $p$?
- 2,907
4
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1
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194
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Structure of $S$-units of norm $\pm1$ in a real quadratic field
Let $K = \mathbb{Q}(\sqrt{d})$ be a real quadratic field, with ring of integers $\mathcal{O}$. Let $n\in\mathbb{Z}$ be an integer, and let $S$ denote the primes of $K$ which divide $n$. Let $\mathcal{...
- 7,527
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3
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348
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The rank of elliptic curves and related quadratic twists
Let $E/\mathbb{Q}$ be an elliptic curve, and let $k_1, k_2$ be square-free integers. Can anything be said about the related elliptic curves
$$\displaystyle E/\mathbb{Q}, E^{(k_1)}/\mathbb{Q}, E^{(k_2)}...
- 26.9k
0
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2
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215
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Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
0
votes
1
answer
170
views
Integral closure in the algebraic closure of $p$-adic numbers
Let $p$ be a prime number and let $\overline{\mathbb{Q}}_p$ be a fixed algebraic closure of the $p$-adic numbers $\mathbb{Q}_p$. It is well know that the ring of integers of $\mathbb{Q}_p$ is the ring ...
- 367
2
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1
answer
128
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Conductor of the Hecke character- power residue symbol
The power residue symbol is the multiplicative character $\bigl(\frac a \cdot\bigr): \mathcal I_\mathfrak m\to \mu_n$ that satistfies $\bigl(\frac a {\mathfrak p}\bigr)\equiv a^{\frac{N\mathfrak p-1}{...
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1
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0
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122
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Rational points on an elliptic curve the denominator of x is a square
Let $f \in \mathbb Q[x]$ be a squarefree cubic polynomial with nonzero constant coefficient and consider the elliptic curve $E : y^2 = f(x)$.
Define $E(\mathbb Q)' \subseteq E(\mathbb Q)$ as
$$\left \...
- 2,224
2
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1
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167
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Difference between smooth and continuous $\ell$-adic representations
I am studying the book "The local Langlands conjecture for GL$_2$" scribed by Bushnell and Henniart. When they talk about $\ell$-adic representations, they assert without explanation that ...
- 367
2
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1
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313
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On the Diophantine equation $2^a3^b + 4^c5^d = 2^b3^a + 4^d5^c$ [closed]
In this year's team selection test to the international mathematical Olympiad in Japan, a interesting diophantine equation appeared as the final problem of the second day.
Find all quadruples $(a,b,c,...
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116
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Restriction of the local Artin map on the valuation ring of a local field
Let $F$ be a local field, in particular a finite extension of $\mathbb{Q}_p$ for some prime $p$ and let $Art_{L}: L^\times \to Gal(F^{ab}/F)$ be its local Artin map. We know that if $L/F$ is a finite ...
- 367
1
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0
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159
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Property of a sequence on $\mathbb Q[\sqrt m~]$
Given that $a_{1} = \sqrt m$ in which $m$ is a integer that is not the square of any integer. And $$a_{n+1}=\frac{[a_{n}]}{\{a_{n}\}}$$where $[~ ]$ and $\{~ \}$ respectively represent the integer part ...
- 11
1
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0
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71
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Relation between the field and $\mathbb{Z}$-algebra generated by eigenvalues of modular form
Cross-posted from MSE: https://math.stackexchange.com/questions/4944262/relation-between-the-field-and-mathbbz-algebra-generated-by-eigenvalues-of
Let $f$ be a cusp form of weight $k\in\mathbb{Z}$ for ...
- 367
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1
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171
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Proof that the infinity norm of basis matrix is greater than or equal to 1
Let $K$ be an algebraic number field of degree $h,$ and let $\beta_1, \ldots , \beta_h$ be an integer basis, so that every integer in $K$ has the unique representation $a_1\beta_1 + \ldots + a_h\...
- 47
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0
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100
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Algebraic degrees of $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + \sqrt{-n}}{2}\right)$ and class numbers of $Q(\sqrt{-n})$
Let $n\in\mathbb N$ be squarefree. Denote by $h(n)$ the class number of $Q(\sqrt{-n})$ and by $d_1(n)$ and $d_2(n)$ the degrees of the algebraic numbers $j\left(\sqrt{-n}\right)$ and $j\left(\frac{1 + ...
- 13.4k
2
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1
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141
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Counting roots of several polynomials modulo a prime
Let $f_1,f_2,\dots,f_k \in \mathbb{Z}[X]$ be a $k$-tuple of polynomials in one variable with integer coefficients. For a prime $p$, let $r_i(p)$ denote the number of roots of (the reduction modulo $p$ ...
- 1,914
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0
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173
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How to factorize the ideal (2) in biquadratic number fields?
Consider the biquadratic number field $K=\mathbb{Q}(\sqrt{n}, \sqrt{m})$, for some square-free integers. My question is how to factorize the ideal $(2)$ in the ring $\mathcal{O}_K$.
There is no hope ...
- 59
14
votes
2
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683
views
Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$?
Can four integer numbers $x$, $y$, $x-y$, $x+y$ be powerful numbers where $\gcd(x,y)=1$ ?
- 1,776
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0
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92
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Gauss sums over number field, residue a prime ideal
Let $\omega$ be a nonzero element of the number field $K$ and write $\mathfrak{a}$ for its denominator.
Hecke defined Gauss sums by
$$C(\omega) = \sum_{x \text{ mod } \mathfrak{a}} \exp(\mathrm{tr}(x^...
- 133
5
votes
2
answers
726
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Powers of Gaussian primes are NOT collinear
Let $p \equiv 1 \pmod{4}$ and let $\pi \in \mathbb{Z}[i]$ be a prime lying above $p$. Let $a$ and $b$ be positive integers.
Is it true that $1, \pi^a, \pi^b$ are collinear in the complex plane if and ...
- 51
2
votes
0
answers
125
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Is it true that all algebraic values of the $j$-invariant have a real Galois conjugate?
This is a follow-up of my recent question on real values of the $j$-invariant. It has had a partial response so far, establishing that any "reality root" $n$ is indeed rational. Now it ...
- 13.4k
0
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0
answers
93
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Existence of maximal totally ramified subextension
Suppose $K/\mathbb Q_p$ is a finite extension, with ramification index $e$ and inertia index $f$. I want to ask whether there exists a subextension $L/\mathbb Q_p$ totally ramified of degree $e$?
- 775
9
votes
2
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1k
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Sets of algebraic integers whose differences are units
Fix a natural number $n\ge2$. Can we find $n$ algebraic integers $a_1,\dots,a_n$ in the field of complex numbers such that $a_i-a_j$ is a unit for all $i\ne j$?
- 1,638
4
votes
1
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222
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Cohen-Lenstra heuristics for ideal class groups over non-abelian extensions
So I'm curious about how the Cohen-Lenstra heuristics on ideal class groups might work over non-abelian extensions. Specifically:
Fix a finite group $G$ and suppose I have a family of Galois ...
- 41