The algebraic-number-theory tag has no wiki summary.

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### Class numbers of orders

Consider an order $R$ in a number field $L$. Let $C_R$ be the set of $R$-fractional ideals modulo $L^\times$. Let $O$ be the maximal order in $L$, and $C_O$ be the class group of $O$.
My question: ...

**2**

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**0**answers

148 views

### Reference request for a basic result on relative differents & discriminants

I am looking for a better reference for the results in this extremely short and elementary paper:
Tôyama, Hiraku,
`A note on the different of the composed field',
Kōdai Math. Sem. Rep. 7 (1955), ...

**8**

votes

**2**answers

407 views

### Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?
Are there infinitely many? Finitely many? ...

**8**

votes

**1**answer

429 views

### Fundamental units of imaginary quartic fields

Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...

**3**

votes

**1**answer

243 views

### $\ell$-conductor of a two-dimensional $\ell$-adic Galois representation

Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...

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**0**answers

77 views

### Divisor bounds of ideals in number fields

Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers).
Denote by $d(I)$ the number of ideals that divide $I$.
So if $I= \prod_{i=1}^k p_i^{e_i}$ is the ...

**3**

votes

**1**answer

124 views

### How to estimate a local hilbert samuel funcion

Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...

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**1**answer

475 views

### Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...

**2**

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**2**answers

249 views

### On Cubic Non-Residues Modulo a Prime [closed]

What is a good test for identifying cubic non-residues/residues and higher power non-residues/residues modulo a prime $R$ in terms of computational complexity?
Given $M$ and $N$, is there a good way ...

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**0**answers

126 views

### What is the real subring of a ring of cyclotomic integers?

I am looking at tilings whose vertices lie in a ring of cyclotomic integers. These tilings are of interest as they can have interesting scaling properties or be substitution tilings. Interesting ...

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**1**answer

1k views

### Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?

From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...

**16**

votes

**1**answer

459 views

### Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?

By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!

**0**

votes

**1**answer

234 views

### For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...

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vote

**1**answer

158 views

### Lower Degree Elements in an Algebraic Number Field

Fix an algebraic integer $\alpha$ of degree $n$
such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields.
(We can assume $K$ is Galois with non-simple Galois group.)
This ...

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votes

**0**answers

101 views

### classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...

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**2**answers

684 views

### Quintic polynomial solution by Jacobi Theta function.

Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...

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**0**answers

83 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 ...

**8**

votes

**1**answer

324 views

### Extensions of Galois representations

Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ...

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votes

**5**answers

2k views

### If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous and amazingly tricky proof says that if we ...

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votes

**1**answer

852 views

### Questions about the proof of Stickelberger's theorem on discriminants

I was going through the proof of Stickelberger's theorem about discriminants in the book 'Algebraic Number Theory' by Richard A. Mollin, and I am having some problems in understanding the proof. I ...

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votes

**0**answers

418 views

### Simplifying an algebraic integer expression

I have an expression where the variables are algebraic integers:
$p4 = \frac{p12 - p41 \cdot p21}{p22}$
p12 is degree 48 and p22 is most likely degree 48 too. p41 is degree 32 and p21 is degree 24. I ...

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**0**answers

969 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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**0**answers

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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votes

**1**answer

137 views

### ramification of discrete valuation field

Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb ...

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84 views

### points in $V(\bar K \otimes_{\bar Q} \bar L)$ rational over tensor product of fields

Let V be a variety over a number field, and let K and L be two algebraically closed
What is known about the points of $V(\bar K \otimes_{\bar Q} \bar L )$ ?
Are there results claiming that points in ...

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**2**answers

763 views

### how to visualize the class number of an imaginary quadratic field?

Let me detail the title of the question. I'm trying to give students an intuition of what the class number is.
Let $K=\mathbb{Q}(\sqrt{-d})$, with $d>0$ a square-free integer, be a quadratic ...

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**0**answers

122 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 ...

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**1**answer

261 views

### Inertia subgroup in the ordinary reduction case when $p=2$

Dear MO,
Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...

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votes

**1**answer

495 views

### Numbers integrally represented by a ternary cubic form

Given integers $a,b,c,$ and cubic form
$$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$
$$ f(a,b,c) =
\det \left( \begin{array}{ccc}
a & b ...

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288 views

### Computing certain class numbers modulo 4

Let $p \equiv 5 \pmod{8}, q \equiv 7 \pmod{8}$ be primes and $N = pq$. I want to show that the class number $n$ of $\mathbb{Q}(\sqrt{-N})$ satisfies $n \equiv 2 \pmod{4}$ if $\left(\frac{q}{p}\right) ...

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**1**answer

253 views

### local field and number field

Let $K$ be a local field (locally compact topological field) of characteristic zero.
Is it true that $K$ is isomorphic to the completion of a number field
under some valuations?
If yes, then how to ...

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**1**answer

760 views

### Principal maximal ideals in Z[x]/(F)

Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain $\mathbb{Z}[x]/(F)$ ...

**1**

vote

**1**answer

416 views

### How can we understand Baker's theorem about transcendence ?

We know that for algebraic $e^{it\pi}$, $t$ can not be algebraic irrational by
Baker's theorem, but his proof is analytic; is there some algebraic understanding for such fact? If $t$ is ...

**0**

votes

**1**answer

308 views

### If e^itπ is algebraic , is $t$ a rational number. [closed]

I have a elementary question:If e^itπ is algebraic , is $t$ a rational number.
I do not know whether it is right

**6**

votes

**1**answer

340 views

### Rational points on surfaces of general type

The weak Lang conjecture asserts that rational points on a variety of general type defined over $\mathbb{Q}$ are not Zariski dense (same replacing $\mathbb{Q}$ with a number field). This one is proved ...

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**2**answers

622 views

### Exercise in Milne's CFT notes

On page 156 of Milne's Class field theory notes available online here, he claims that the Hilbert class field of $K = \mathbb Q(\sqrt{-6})$ is the splitting field of $x^2+3$ but I don't believe so.
...

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vote

**1**answer

200 views

### Functional equations of zeta functions over global fields

The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in ...

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**0**answers

44 views

### on degree zero elements in adelic groups

Let $G$ a split connected reductive group and $G(\mathbb{A})$ his points in the ring of adeles.
We have a degree map $G(\mathbb{A})\rightarrow X_{*}(Z)$ where $Z$ is the center of $G$.
Let ...

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votes

**1**answer

169 views

### An expression for the function $f_e$ that appears in the Weil Pairing

Let $K$ be a local field and $E/K$ an elliptic curve such that the set of $N$-torsion points, $E[N]$, is contained in $E(K)$. For $e$ in $E[N]$, I am interested in finding and expression for the ...

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480 views

### on the set of numbers generated by integer linear combination of two real numbers.

Let $b > a > 0$ be two real numbers. I am interested in the set of numbers
$X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$.
What ...

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**4**answers

434 views

### Decomposition of primes in Galois closures of number fields

Let $L/K$ be an extension of number fields, and $M/K$ the Galois closure of $L/K$ (everything happens inside a suitably large characteristic zero field $\Omega$). Let $p$ be a discrete prime of $K$.
...

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157 views

### linear independence of orbits via a set of transformations in char p

Let $T_1, \ldots, T_n \in GL(n,\mathbb{F}_p)$. Suppose for all $\vec{v} \in \mathbb{F}_p^n$ we have $\det (T_1 \vec{v}, T_2 \vec{v}, \ldots, T_n \vec{v}) = 0$. Now, let $k$ be a finite extension of ...

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**1**answer

1k views

### Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer.
As we all know, ...

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249 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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246 views

### galois cohomology over finite field

Let $X$ a smooth projective geometrically connected curve over a finite field $k$. Let $J$ a smooth commutative group scheme over $X$ and $F$ the function field of $X$.
Do we have a formula to ...

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403 views

### On Weil's characters of type (A)

In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite ...

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237 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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268 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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### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

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**1**answer

526 views

### Décomposition des nombres premiers dans des extensions non abéliennes

Gauß famously determined the cubic character of $2$ in his Disquisitiones : $2$ is a cube modulo a prime number $p\equiv1\mod3$ if and only if $p=x^2+27y^2$ for some $x,y\in\mathbf{Z}$. This implies ...