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

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### Results for resolution of equations in polynomial ring

Is there any reference for resoluttion of equations in a polynomial ring, such as $x^2+y^2=z^2$ in $\mathbb{C}[t]$? Thanks!

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### Multiplicativity of the Ideal Norm

I asked this question on StackExchange twice, and was never able to get an answer. This is not quite a research-level question, but it is sufficiently difficult and interesting from a pedagogical ...

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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 ...

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### A question on linear forms

Let $K/\mathbb{Q}$ be a fixed number field of degree $n$, and let $\sigma_1, \cdots, \sigma_n$ denote the distinct embeddings of $K$ into $\mathbb{C}$. Suppose that $\alpha_1, \cdots, \alpha_{r}$ are ...

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### 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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### 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 ...

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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?

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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, Disquisitiones ...

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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: ...

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### Why do noncocompact arithmetic Kleinian groups have quadratic trace fields?

I realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find ...

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### On the conductor of the Groessencharacter of a CM elliptic curve

Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that ...

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### Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...

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### 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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### 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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### What is the explicit eigenvalues of Hilbert modular forms?

Let $F$ be a totally real number field and let $I$ denote the set of embeddings
$\tau:F\to \mathbb{R}.$ Let $k=(k_\tau)\in\mathbb{Z}^I_{>0}$ and suppose all the $k_\tau$'s have the same parity. Let ...

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### Conjecture on irrational algebraic numbers

Conjecture:
For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.
Questions:
Has this conjecture been ...

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### closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?

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### Chinese Remainder Theorem backwards

I have the following situation, that is much alike the Chinese Remainder Theorem. Let $\phi_d(\alpha)$ be the $d^{th}$ cyclotomic polynomial in the variable $\alpha$ (I'm not specifying the ...

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### 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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### Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$

Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, ...

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### Finding a suitable number

Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, ...

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### Unique factorization in a quadratic extension of $Z_p$ [closed]

Let $\mathbf{Z}_{(p)}$ denote the ring of all rational numbers whose denominators in lowest terms are not divisible by the integer prime $p$. (This is normally described as the localization at $p$.) ...

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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 ...

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### Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4

In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...

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### Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:
Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected ...

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### Positive primes represented by indefinite binary quadratic form

Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, ...

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### Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree.
Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$?
...

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### Where can I find the article of A. Borel: “Values of zeta-functions at integers, cohomology and polylogarithms”? [closed]

Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.

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### Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation.
Since I essentially need $n\le 4$, I think that I can show it ...

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### Neukirch's papers and theorem

Have any of Neukirch's papers on anabelian geometry been translated? I'm mostly interested in:
Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper (1969)
Kennzeichnung der ...

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### Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$

I know that number fields have been the object of many statistical experiments.
Is there some kind of heuristics for the following?
Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic ...

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### Examples of specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$
indeterminates. The $h^{th}$elementary symmetric polynomial is the
sum of all monomials with $h$ factors
\begin{eqnarray*}
...

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### GL(2) Local Langlands and Artin's L-function

The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome.
In light the local Langlands correspondence, what ...

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### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

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### unit group of biquadratic fields

In the unit group of a real biquadratic field, what is the index of the product of the unit groups of its three quadratic subfields?
Is the index 1 if discriminant of these three subfields are always ...

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### Orders of the conjugates of an algebraic prime number in its residue field

Of interest to me is the following question (it would be nice to find out what is known in its direction):
Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime ...

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### Reference for the fact that $SL_n(O_K)$ surjects onto $SL_n(O_K/I)$ for any ideal I

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $I \subset \mathcal{O}_K$ be a nonzero proper ideal. It is not hard to see that the map ...

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### Quotients of number rings IZ[zeta_l]

Let $l=p^r$ a prime power and $\zeta$ a primitive l-th root of unity. It is classical result, that $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon\in\mathbb{Z}[\zeta]$ for a unit $\epsilon$.
It should be a ...

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### Programs to find class number of bi-quadratic fields

I'm interested in studying ideal class group structure for bi-quadratic fields.
The problem I'm facing is the lack of examples.
Is there any program where I can find the class number of a bi-quadratic ...

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### Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...

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### Extensions on Higher-dimensional local fields

I have the following question:
Let $M/L$ b a finite extension of n-dimensional local fields and $t_1,\dots, t_n$ a system of local parameters of $L$ with valuation $v$. Let us fix an $1\leq i \leq ...

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### Valuations in Higher-dimensional local fields

I have the following question which I believ should be true but I would like to have a different opinion about it:
Let $M/L$ is a finite Galois extension of $n$-dimensional local fields and ...

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### primitive polynomial on $\mathbb{F}_2[x]$

For some reason I need some primitive polynomial $f$ on $\mathbb{F}_2[x]$ where $\deg f \in [1,10^4]$. (Especially for $\deg f = 10\pm \epsilon, 10^2\pm \epsilon, 10^3\pm \epsilon, 10^4 \pm ...