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### Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...

**24**

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734 views

### Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$,
and define the Fekete polynomials
$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$
Define
$$ f_N(X) = ...

**20**

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942 views

### Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$.
Question: Let $p$ be an ...

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681 views

### Most “natural” proof of the existence of Hilbert class fields

Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...

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758 views

### Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...

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244 views

### 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 ...

**10**

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501 views

### Class groups in dihedral extensions - some sort of Spiegelungssatz?

Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ...

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514 views

### Points of bounded height in a number field

Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...

**9**

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291 views

### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

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720 views

### Dissecting trapezoids into triangles of equal area

[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark]
The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...

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145 views

### 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 ...

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336 views

### Effective lower bound for class numbers of cyclotomic fields

Let $K=\mathbb{Q}(\mu_p)$ with class number $h=h^+h^-$, where as usual $h^+$ is the class number of the maximal real subfield of $K$. My question is whether there is an effective lower bound for $h$ ...

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158 views

### Computation of relative class groups

In the explicit construction of Hilbert $p$-class fields of a number field $K$ it is not so much the class group of $K$ or that of $L = K(\zeta_p)$ that is needed but the relative class group of the ...

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256 views

### 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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269 views

### Does Hasse-Minkowski help to produce nontrivial rational solutions?

Consider a quadratic form over $\mathbb{Q}$, say, a diagonal one in three variables
$$
F(X, Y,Z) = a · X^2 + b · Y^2 − c · Z^2
$$
with positive integers $a,b,c$. Then $F(X,Y,Z)=0$ has a non-trivial ...

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75 views

### Is the equidissection spectrum closed under addition?

If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area?
(I'm editing after realizing that my conjecture that a ...

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247 views

### Do the Adeles Split?

I asked this question about a week ago here http://math.stackexchange.com/questions/288955/splitting-the-exact-sequence-of-the-idele-class-group, but got no answer so I thought I'd aske here and see ...

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309 views

### Tameness criterion in the reducible case

Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...

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392 views

### The character of a separable degree-$p$ extension of a local field of residual characteristic $p$ ?

Let $p$ be a prime number and $F$ a finite extension of ${\mathbf Q}_p$ or of ${\mathbf F}_p((t))$. I'm going to define a natural map from the set ${\mathcal S}_p(F)$ of degree-$p$ separable ...

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423 views

### ideal classes in quadratic number fields

Let $m$ be a squarefree odd integer that can be written as a sum of two squares, and let $K = {\mathbb Q}(\sqrt{m}\,)$ be a real quadratic number field with fundamental unit $\varepsilon$.
Let $m = ...

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220 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 ...

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393 views

### Is formula valid for relating $\pi$ with ALL of its OEIS A002485(n)/A002486(n) convergents?

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ?
$$(-1)^n\cdot(\pi ...

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76 views

### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...

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255 views

### Differences in tree picture of ${\bf Q}_p$, $\overline{{\bf Q}_p}$, ${\bf C}_p$, $\Omega_p$

I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value ...

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144 views

### How to construct Weil numbers in a given CM quartic field?

Let $L$ be a CM field of degree $4$ over the rationals, and let $p$ be a prime number. If $q$ is a power of $p$, I would like to know if it is possible to characterize (in some way) all Weil ${\bf ...

**6**

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622 views

### Automorphisms of local fields

It is an amusing coincidence (at least it appears to be a coincidence to me) that any completion of the field $\mathbb{Q}$ has trivial automorphism group as an abstract field, i.e. when ignoring the ...

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264 views

### maximal abelian extension of quadratic extension of $\mathbb Q_p$

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb ...

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190 views

### Generating congruence subgroups of SL_n over totally imaginary number rings

Fix some $n \geq 3$. Let $k$ be an algebraic number field with ring of integers $\mathcal{O}$ and let $\alpha$ be an ideal of $\mathcal{O}$. Define $\text{SL}_n(\mathcal{O},\alpha)$ to be the ...

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158 views

### Salem and Perron polynomials

If $P(t)\in \mathbb{Z}[t]$ is a polynomial, let $d$ be its degree and let $P_{*}(t)$ denote its reciprocal polynomial, i.e. $P_{*}(t) := t^d\, P(1/t)$.
Let $Q_n(t) \in \mathbb{Z}[t]$ be a polynomial ...

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149 views

### On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity.
...

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636 views

### Reference for the Odd Dihedral Case of Artin's conjecture

The example that Matt Emerton cited here:
prompted me to become interested in how one proves that odd two dimensional dihedral Galois representation are modular. This is the first case of the strong ...

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72 views

### Iwasawa theory: Do these $\mu$-invariants of a number field coincide?

Let $k$ be a number field and let $k_\infty$ be the cyclotomic $\mathbf{Z}_p$-extension of $k$. Put $\Gamma=G(k_\infty/k)\cong \mathbf{Z}_p$, $\Lambda=\mathbf{Z}_p[[\Gamma]]$. Let $S$ be a finite set ...

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137 views

### Continuity of the Hilbert pairing

I would like to know if the Kummer pairing (or the analogue of the Hilbert Symbol) for a one dimensional group defined over the ring of integers of a higher-dimensional local field is continuous (with ...

**4**

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168 views

### When is a Number Ring generated by its Norm-1 elements?

In exercise 1.2.11. of Bump's Automorphic Forms and Representations book, he deals with real quadratic fields $K$ in which $\mathcal{O}_K$ is generated (as a ring) by its norm-1 units. In this case ...

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231 views

### What else does the Tate-Nakayama lemma tell us about class field theory?

Doing class field theory from the point of view of class formations, it is my understanding that to get the artin reciprocity map, one does this by inverting an isomorphism which is given by the ...

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183 views

### Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?

Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...

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259 views

### On Stickelberger's Theorem over function fields

Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields).
Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...

**4**

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221 views

### A natural mod-n representation of the automorphisms of an elliptic curve

Let $E$ be an elliptic curve over a field of characteristic $p$, and let $n$ be an integer coprime to $p$. Then $E[n]$, the kernel of multiplication by $n$ on $E$, is (etale-locally) isomorphic to ...

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### Detecting linear dependence on multiplicative groups

Let G = $\mathbb{G}_m^2/\mathbb{Q}$ and let $\Gamma \subseteq G(\mathbb{Q})$ be a free abelian group of rank 2. Assume that the set of primes $p$ for which $\Gamma \mod p$ is cyclic has positive ...

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266 views

### What is the shape of the zeta function of a singular hypersurface?

So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$.
Assume that
(a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible,
(b) and that ...

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320 views

### Why a certain “Hecke polynomial” is equal to a certain “Galois polynomial”?

Let $f$ be a modular form of weight $k>0$ on $\Gamma_1(N)$ that is an eigenvector for all the Hecke operators $T_\ell$, for $\ell$ prime, and with Nebentype $\epsilon$. If $\ell$ is a prime not ...

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268 views

### What to call the following variant of tame ramification

Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...

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119 views

### Computing local volumes : the case of Hecke p-adic subgroups

I am quite interested in knowing how to compute some volumes of groups defined on local fields $K$, mainly in order to evaluate the identity term in trace formulas. It is something well done in the ...

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246 views

### Independent units in pure number fields $\mathbb{Q}(\sqrt[p]{t})$

Theorem: In this paper of Frei and Levesque, they correct the proof of a result of Halter-Koch and Stender:
Define the real pure algebraic number field $\mathbb{K}=\mathbb{Q}(\omega_n)$ for ...

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139 views

### Is Eisenstein's map from binary cubic forms really injective when $\Delta > 0$?

The four binary cubic forms, homogeneous cubics of the form $C_j(x,y) = a x^3 + b x^2 y + c x y^2+d y^3 = (a,b,c,d)$,
$C_1 = (1,3,-15,-23)$, $C_2 = (1,3,-51,-203)$, $C_3 = (1,3,-33,-113)$, $C_4 = ...

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188 views

### Euclidean real quadratic fields

It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic ...

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243 views

### Inequalities in paper by Jean Bourgain

The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053
Specifically, I can't derive the following inequality in (1.20):
\begin{equation}
...

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121 views

### Diophantine approximations by norms of quadratic irrrationalities

The following problem came up on a mailing list that I subscribe to:
If $\alpha$ is irrational we can find (using continued fractions) infinitely many rational fractions $p/q$ such that $|q \alpha - ...

**3**

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88 views

### Decompositions of representations of pro-p groups

Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...

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119 views

### P-adic Weierstrass Lemma for several variables

The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...