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25
votes
0answers
433 views

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
votes
0answers
714 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
votes
0answers
929 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 ...
16
votes
0answers
662 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 $K$ ...
13
votes
0answers
744 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 ...
10
votes
0answers
490 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 $p$ ...
10
votes
0answers
508 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
votes
0answers
701 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 ...
8
votes
0answers
334 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 $h$ ...
8
votes
0answers
155 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 ...
7
votes
0answers
254 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 ...
7
votes
0answers
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 ...
7
votes
0answers
245 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 ...
7
votes
0answers
303 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 ...
7
votes
0answers
391 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 ...
7
votes
0answers
419 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 = ...
6
votes
0answers
212 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 ...
6
votes
0answers
391 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 ...
6
votes
0answers
73 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 ...
6
votes
0answers
246 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 ...
6
votes
0answers
138 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
votes
0answers
606 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 ...
5
votes
0answers
252 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 ...
5
votes
0answers
187 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 ...
5
votes
0answers
150 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 ...
5
votes
0answers
143 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. ...
5
votes
0answers
260 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 ...
5
votes
0answers
629 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 ...
4
votes
0answers
65 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 ...
4
votes
0answers
135 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
votes
0answers
167 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 ...
4
votes
0answers
229 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 ...
4
votes
0answers
178 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 ...
4
votes
0answers
253 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
votes
0answers
218 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 ...
4
votes
0answers
115 views

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 ...
4
votes
0answers
265 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 ...
4
votes
0answers
317 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 ...
4
votes
0answers
267 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, ...
3
votes
0answers
242 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 ...
3
votes
0answers
134 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 = ...
3
votes
0answers
179 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 ...
3
votes
0answers
231 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} ...
3
votes
0answers
119 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
votes
0answers
85 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 $P$, ...
3
votes
0answers
118 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 ...
3
votes
0answers
137 views

Modified radical group of a Kummer extension

If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$ --- let's also assume $K/k$ is not generated by $p$-power roots of unity), I ...
3
votes
0answers
416 views

A proof of a theorem on the different in algebraic number fields

I asked this question in Stack Exchange, here, but I got no answer so far. I don't know any modern book on algebraic number theory which states the following theorem, let alone its proof except ...
3
votes
0answers
337 views

Cyclotomic fields and singular moduli

Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?
3
votes
0answers
271 views

idelic closures of units of number fields

Let $K$ be a number field, $\mathcal O _K^\times$ its group of integral units and $\mathcal O _{K,+} ^\times$ its group of totally positive units. Denote further by $\widehat{\mathcal O }_K^\times$ ...