**21**

votes

**0**answers

381 views

+300

### Simple argument regarding sums of two units in a number field?

I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...

**66**

votes

**4**answers

8k views

### How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...

**3**

votes

**1**answer

89 views

### Bibliography suggestion for Kummer theory

I already posted a question about a sum involving the degree of a Kummer extension.
Now I'm interested in a more specific fact about Kummer extensions.
From Hooley's paper "On Artin's conjecture", we ...

**2**

votes

**1**answer

114 views

### Doubt concerning a sum involving Kummer extension degrees

I'd like to estimate the following sum
$$
\sum_{n\leq x}\frac1{k_n}\;,\qquad x\rightarrow \infty\;,
$$
where
$k_n=[\mathbb{Q}(\zeta_n,a^{1/n}):\mathbb{Q}]$
is the degree of a Kummer extension for a ...

**8**

votes

**5**answers

3k views

### Has anyone found an error in an early version of Neukirch?

I remember a friend in graduate school throwing an early edition of Jurgen Neukirch's Algebraic Number Theory book against a wall (so hard that it split the binding) after he had worked for a number ...

**2**

votes

**0**answers

102 views

### Maximal abelian extension and tamely ramify extension

Let $K$ be a number field and $v$ a finite place of $K$ lying above a prime number $p$. Assume further that $v$ is unramified over $p$. If $K^{ab}$ is the maximal abelian extension let $w$ be a ...

**1**

vote

**0**answers

94 views

### Completion of a finite field extension is also finite?

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completion of $(L,w)$, $(K.v)$. Is the field extension $L_w/K_v$ finite?
For nonarchemidean ...

**9**

votes

**1**answer

389 views

### Elementary proof of a special case of Chebotarev's density theorem

A special case of Cheboratev's density theorem states that, for $K/\mathbb{Q}$ a Galois number field of degree $n$, then the rational primes that split completely in $K$ have density $1/n$.
Is there ...

**7**

votes

**0**answers

158 views

### Geometric meaning of conductor

Supppose $L/K$ is a finite extension, choose $\theta \in O_L$ such that $L=K(\theta)$. We define the conductor of ring $O_K[\theta]$ to be an ideal of $O_L$, namely: $F=\{\alpha\in O_L|\alpha\cdot ...

**2**

votes

**0**answers

122 views

### calculation in a group ring

I have some problems with the verification of the third equation in Lemma 1 in this paper.
First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...

**15**

votes

**3**answers

2k views

### sum of squares in ring of integers

Lagrange proved that every (positive) rational integer is a sum of 4 squares.
Are there general results like this for ring of integers of a number field? Is this class field theory?
Explicity, ...

**2**

votes

**1**answer

186 views

### Irreducibility of Faulhaber-like Polynomials over $\mathbb Q[x]$

Motivation: Inspired by the famous Faulhaber polynomials $F_k(N)=\displaystyle\sum_{n=0}^Nn^k,$ I decided to study their alternating versions, $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$.
For ...

**5**

votes

**0**answers

84 views

### On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...

**6**

votes

**2**answers

197 views

### Number of representations as sums of squares in rings of integers of number fields

Let $K$ be some number field, $\mathcal O_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O_K$, and consider the quantity $r_{n,K}(\alpha)$, which denotes ...

**10**

votes

**1**answer

430 views

### A set of generators for $\bar{\mathbb{Q}}$

Two questions:
Does there exist a sequence $\alpha_1,\alpha_2,...$ of algebraic numbers with degrees $d_1,d_2,...$ s.t. for each $i$, $d_i|d_{i+1}$ and $\alpha_i= p_i(\alpha_{i+1})$ with $p_i$ a ...

**8**

votes

**0**answers

86 views

### Nondegenerate quadratic form $Q$ in $n$ variables with coefficients in $F_p$, cardinality of $\{x \in (\mathbb{F}_p)^n : Q(x) = 0\}$ [migrated]

There is a book on algebraic number theory I am reading that shows the following theorem.
Let $Q$ be a nondegenerate quadratic form in $n$ variables with coefficients in $F_p$ ($p \neq 2$). ...

**8**

votes

**0**answers

151 views

### What is the relationship between the conductor of an order and the conductor of a number field extension?

What is the relationship between the conductor $\mathfrak{f}_{\mathcal{o}}$ of an order $\mathcal{o}\subset \mathcal{O}_K$ and the conductor $\mathfrak{f}_{L/K}$ of a field extension in the classical ...

**6**

votes

**1**answer

176 views

### Can a product of conjugates be a Pisot number again?

Let $p(X) \in \mathbb{Z}[X]$ be an irreducible polynomial, and let $\alpha_1 \dots, \alpha_n$ be its roots in $\mathbb{C}$. Suppose that $\alpha_1$ is a Pisot number, that is, $\alpha_1 \in ...

**0**

votes

**0**answers

149 views

### Writing integers in ring of integers of number fields

Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of ...

**6**

votes

**2**answers

495 views

### On bounds for idoneal integer

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...

**9**

votes

**0**answers

138 views

### The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

Recall the notion of a "nearly holomorphic modular form" introduced by Shimura:
A function $f : \mathfrak h \to \mathbb C$ is said to be nearly
holomorphic of level $\Gamma_1(N)$, weight $k$ and ...

**2**

votes

**0**answers

145 views

### For $K/E$ a number fields extension and $F/E$ a finite Galois extension, how is the ramification in $F\cdot K/K$ related to the one in $F/E$?

Studying class field theory, I have come across the following Proposition:
Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with ...

**2**

votes

**2**answers

154 views

### Counting number of $2\times 2$ unimodular matrices of particular type

From set of numbers from $\Bbb S=\{0,1,\dots,m\}$, how many distinct $3\times 3$ unimodular matrices parametrized by $(a,b,c,d,e,f)\in\Bbb S^6$ of following type can one form?
\begin{bmatrix}
a^2 ...

**2**

votes

**0**answers

64 views

### Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...

**8**

votes

**2**answers

520 views

### Does the Galois group of a Pisot polynomial contain the alternating group?

Let $n \in \mathbb{N}$, and let $p(X) \in \mathbb{Z}[X]$ be a monic polynomial of degree $n$. Suppose that exactly one complex root of $p$ is of modulus $> 1$, and that the remaining $n-1$ roots of ...

**2**

votes

**2**answers

201 views

### Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...

**2**

votes

**1**answer

177 views

### On Pell's equation

A post was made (Reduction from factoring to solving Pell equation) seeking clarification to solving $$x^2-Dy^2=1$$ to factoring when $D>0$. An answer was posted stating that to factor $N$, it ...

**2**

votes

**1**answer

106 views

### Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then:
($\ast)$ Let $E/K$ is a finite separable field extension, then the ...

**5**

votes

**2**answers

325 views

### Finding the inertia group

Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$.
What is the isomorphism class of the inertia group $I_p$, ...

**4**

votes

**0**answers

150 views

### Expressing every algebraic number using roots of trinomials?

This question is a continuation of Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m ...

**5**

votes

**1**answer

386 views

### Is every polynomial a factor of a trinomial?

We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each ...

**1**

vote

**0**answers

98 views

### In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$.
Let $0$ be one of the origins of ...

**4**

votes

**0**answers

167 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 ...

**7**

votes

**3**answers

619 views

### Ranks of elliptic curves depend only on the field?

Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?

**23**

votes

**2**answers

2k views

### How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...

**1**

vote

**0**answers

149 views

### An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?

**9**

votes

**0**answers

93 views

### Meromorphic functions on $U^2 = T^3 + 1$, cokernel of $O_S \to F_\infty/O_\infty$ [closed]

See here. Crossposted from math.stackexchange since there's no good answer despite $>$ 20 upvotes.
Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of ...

**13**

votes

**2**answers

495 views

### What are Mean Values of Ideal Densities in Galois Extensions?

In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote:
"Classifying extensions according to the Galois group
of their normal closure ...

**9**

votes

**2**answers

598 views

### Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...

**18**

votes

**1**answer

940 views

### Can Eisenstein's lattice point proof of quadratic reciprocity be generalized?

I'm referring to this proof. The key formula ("Eisenstein's Lemma") is
$$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where $u=2,4,\ldots,p-1$}$$
The sum in the exponent ...

**5**

votes

**2**answers

274 views

### Iwasawa's mu-invariant for noncyclotomic $\mathbf{Z}_p$ extensions of cyclotomic fields?

Let $p$ be an odd prime number, $m$ a positive integer with $p\mid m$. Put $k=\mathbf{Q}(\mu_m)$.
(1) Is there any example where certain noncyclotomic $\mathbf{Z}_p$-extension $k_\infty/k$ has ...

**29**

votes

**3**answers

2k views

### On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...

**11**

votes

**2**answers

457 views

### Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

**7**

votes

**2**answers

311 views

### Field of definition of dominant morphisms

Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes ...

**8**

votes

**1**answer

577 views

### Remark 4.23.4 in Hartshorne

Crosspost from math.stackexchange, since it's quite possible I might not get a response there.
Remark 4.23.4 in Chapter IV of Hartshorne's Algebraic Geometry references a paper by Elkies that ...

**9**

votes

**2**answers

443 views

### Intuition behind Kronecker's congruence?

The modular polynomial is defined by$$\Phi_n(X, \tau) = \prod_{\tau} (X - j(\tau)),$$where $j$ is the elliptic modular function and $\tau$ is running through classes of imaginary quadratic integers of ...

**14**

votes

**1**answer

305 views

### Why there are only finitely many $\overline{\mathbb{Q}}$-isomorphism classes of elliptic curves with CM by $\mathcal{O}$?

For someone who does not have a very extensive knowledge of number theory, what is a good intuitive explanation as to why there are only finitely many $\overline{\mathbb{Q}}$ isomorphism classes of ...

**1**

vote

**1**answer

93 views

### Freeness of the group of principal ideals of a number field

This is just a question wondering whether a concrete (enough) result that can be proved using the axiom of choice can be proved without it. The result being that the group of principal ideals $P_K$ of ...

**0**

votes

**1**answer

95 views

### Computation of Hilbert symbol of order 4

We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for ...

**14**

votes

**0**answers

265 views

### Result of Deuring, intuitive way to see it's true/quickest way to prove?

There is the following result of Deuring that goes as follows:
Let $E/L$ be an elliptic curve defined over a number field $L$ with complex multiplication by an order $\mathcal{O}$ in an imaginary ...