Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

0
votes
1answer
52 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
2
votes
0answers
23 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
0
votes
1answer
53 views

Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the rings of $p$-adic integers such that $f\equiv g \...
2
votes
0answers
55 views

A prime number $p$ is ramified in $\mathbb{Q}(\sqrt[p]{a})$.

Let $p$ be an odd prime number and $a\in \mathbb {Z}$ with $\sqrt[p]{a}\notin \mathbb{Z} $. Prove that $p$ is ramified in the number field $\mathbb{Q}(\sqrt[p]{a})$. My idea is to apply Dedekind's ...
7
votes
1answer
1k views

On determining the ring of integers of a cubic number field

I have the following question: Let $\alpha$ be a root of the polynomial $f(x) = x^3-x+1$, and let $K = \mathbb{Q}(\alpha)$. Show that $\mathcal{O}_{K} = \mathbb{Z}[\alpha]$. As I understand it, ...
29
votes
6answers
2k views

Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$

This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. ...
3
votes
2answers
211 views

Ramified prime in cyclotomic extension of a number field

Let $K$ be a number field, $n$ be a positive integer and $\zeta_n$ a primitve $n^{th}$ root of unity. How does one show that if a prime ideal $\mathfrak{p}$ of $K$ is ramified in $K(\zeta_n)$ then ...
2
votes
1answer
32 views

Understanding the notation of a paper

I am reading a paper on Algebraic Number Theory that says If $p$ divides the discriminant of polynomial $f$ $r$ times and there is the factorization into irreducibles $$f(x)\equiv g_1(x)\dots g_r(...
1
vote
1answer
27 views

Existence of units in number fields outside the rings of integers

Let $K/ \mathbb Q$ be a number field with $[K:\mathbb Q]=n$. Using that there exists a prime $p\in \mathbb Z$ which splits completely, that is $p\mathcal O_K=P_1...P_n$ for some distinct primes $P_i$ ...
0
votes
0answers
14 views

$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
1
vote
0answers
47 views

Finding square root of polynomial in extension field (Number field sieve)

I was reading this paper, on page number 29, 2nd paragraph it is written that "take the coefficients of $ \gamma $ modulo q and applying an algorithm for taking square roots in the finite field Z[ $ \...
0
votes
0answers
40 views

Classification of set of algebraic integers in $F$

Let $F$ be a field with $\mathbb{Q} \subseteq F \subseteq \mathbb{C}$, where $F/\mathbb{Q}$ is a finite abelian Galois extension. Then can we classify set of algebraic integers in $F$ ?
2
votes
0answers
33 views

Trace of Witt vectors

Let $\mathbb{F}_p$ be a finite field with $p$ elements, and $\kappa := \mathbb{F}_q$ an extension of $\mathbb{F}_p$ of degree $n$ with $q = p^n$. Then $W_\infty(\kappa)$ is a ring extension of $W_\...
3
votes
1answer
94 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
5
votes
1answer
53 views

Partitioning $\mathbb{P}^1(K)$ via the class group

Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\...
2
votes
1answer
51 views

Representative of the $2$-Sylow subgroup of an ideal class group

Let $C$ be the ideal class group of $\mathbb{Q}(\sqrt{-6})$. I already showed that the ideal $(2,\sqrt{-6})\in C$ is not principal in $\mathbb{Q}(\sqrt{-6})$, but it is principal in $\mathbb{Q}(\sqrt{...
3
votes
0answers
67 views

When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
4
votes
0answers
107 views

Prove that a specific ring of integers is not monogenic

I'm trying to prove that the ring of integers of $K=\mathbb{Q}(\sqrt7, \sqrt13)$ is not of the form $ \mathbb {Z}[a]$ for some $a$. Unfortunately I can not figure out where to start. I tried to ...
2
votes
4answers
151 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
4
votes
1answer
40 views

When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...
5
votes
1answer
87 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
3
votes
1answer
84 views

Ideal which becomes a principal ideal in a higher field extension

I am working on the question of why the ideal $(2,\sqrt{-6})$ is not a principal ideal in $\mathbb{Q}(\sqrt{-6})$, but becomes one in $\mathbb{Q}(\sqrt{-6},\sqrt{2})$. To prove that it is not ...
3
votes
1answer
672 views

How do we find the prime ideals of a ring of integers of a number field?

For example, for $F=\mathbb Q(\sqrt{-5})$ the ring of integers of $F$ is $\mathbb Z[\sqrt{-5}]$ (since $-5\equiv3 \pmod 4$). How can we determine the prime ideals of this ring? Another problem is the ...
1
vote
2answers
40 views

Reference request: Binary quadratic forms

I am currently a first year grad student doing an independent study on topics in algebraic number theory and am currently looking at some of the properties of the polynomial $n^2 + n + A$, where $A \...
3
votes
1answer
60 views

Norm of roots of unity conjugated by Galois automorphisms in CM-fields

Let $(K_n)_{n\geq0}$ be a sequence of CM-fields, so that $K_0\subset K_1\subset\dots$ with $[K_{n+1}:K_n]=p$ for all $n\geq0$. For $n\geq0$ let $W_n$ be the group of the roots of unity in $K_n$. Now ...
0
votes
0answers
32 views

If $p$ is unramified in every subfield of $K$, does it mean $p$ is unramified in $K$?

I am wondering if $p$ being unramified in every subfield of $K$ means $p$ is unramified in $K$. Any hints?
0
votes
1answer
27 views

Class group embedding in coprime extension

Let $L/K$ be an extension of number fields of degree $n$. Assume that the class group of $K$ has order $h$. Prove that if $(h,n)=1$ the map $Cl(K)\rightarrow Cl(L)$, given by $I\rightarrow I\mathcal ...
9
votes
3answers
245 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over $\...
5
votes
0answers
64 views

Show that $\sum_{d\mid f} \varphi(f/d) a^{|d|} \equiv 0 \pmod f$

This equation is correct when $f$ and $a$ are any integers. I want to show that this holds for $f,a\in K[x]$ where $K$ is any finite field. In the equation $\varphi(f)$ is defined as $|(K[x]/(f))^\...
2
votes
1answer
57 views

Literature on fields $\mathbb{Q}(\sqrt[4]{D})$

Can anyone direct me to literature discussing extension fields of the form $K = \mathbb{Q}(\sqrt[4]{D})$ where $D$ is squarefree? I'm particularly interested in results regarding the class and unit ...
6
votes
2answers
68 views

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
2
votes
0answers
39 views

Finding units in quadratic integer rings

I want to find the units in $\mathbb{Z}[\alpha]$, where $\alpha=\frac{1+\sqrt{-11}}{2}$. One can of course use norms to find the units in quadratic integer rings of the form $\mathbb{Z}[\sqrt{D}]$ ...
2
votes
2answers
40 views

How to find the class number of $\mathbb{Q}(\sqrt{-17})$?

I tried to calculate the class number with help of the Minkowski bound of $M \approx 5$. So if an ideal has norm $1$, it is the ring of integers. If it has norm $2$, it is $(2, 1+\sqrt{-17})$, which ...
0
votes
0answers
23 views

Embeddings of $K_v$ in $\mathbb{C}$

Let $K$ be a number field, $v$ a nonarchimedean prime, and $K_v$ the completion of $K$ at $v$. We have the embedding $K \to K_v$, and also $K \to \mathbb{C}$. I have two related questions: Is ...
1
vote
1answer
33 views

construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$?

how might I construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$? Can it be done using pigeonhole principle as with square roots and Pell equation. I had been reading about the Voronoi continued fraction or ...
3
votes
1answer
569 views

How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
1
vote
1answer
29 views

Understanding a proof in Washington's “Cyclotomic Fields”

I'm working through Washington's "Cyclotomic Fields" and having a problem with the proof of Proposition 3.8, which states: Given an abelian group G, there is an everywhere-unramified extension of ...
1
vote
2answers
68 views

Galois principle for ideals

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Determine a necessary and sufficient condition on $L/K$ to ensure that $$\{I\in \text{Id}_L,\text{ such that }\sigma (I)...
7
votes
1answer
224 views

A generalization of Ostrowski's Theorem [closed]

Let $D$ be a Dedekind domain and $|\cdot|$ be a bounded, nontrivial norm on $D.$ Is $|\cdot|$ equivalent to an $I$-adic norm for some maximal ideal $I$ of $D?$
2
votes
1answer
117 views

What is number theory today? [closed]

I try to explaine my problem and I hope do not disturb or annoy; I know that number theory is very vast but essentially it is divided into two parts: analytic number theory and algebraic number ...
3
votes
0answers
47 views

How a complex root $\eta$ of $x^2 + x + A$ affects the ring $\mathbb{Z}[\eta]$

While reading a statement in P. Pollack's Not Always Buried Deep: A Second Course in Elementary Number Theory I came across a statement that seemed obvious and I am wondering if I am oversimplifying ...
10
votes
2answers
247 views

What are the units in $\mathbb{Z}[\root 3 \of 2]$?

I asked Wolfram Alpha to tell me the fundamental unit of $\mathbb{Z}[\root 3 \of 2]$, it replied $1 - \root 3 \of 2$. Then I tried asking it for $(1 - \root 3 \of 2)^n$ for $-5 \leq n \leq 5$. If I ...
0
votes
1answer
413 views

Prime divisors of the conductor of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of ...
1
vote
2answers
60 views

Restriction from subgroup of the Galois group of max. unr, ext. $G(\tilde{K}/\mathbb{Q}_{p})$ to $G(K/\mathbb{Q}_{p})$ is surjective?

This is a question I'm struggling with for some time. Let $K$ be a finite Galois extension of $\mathbb{Q}_{p}$ and let $\tilde{K}$ denote the maximal unramified extension of $K$. We can then ...
3
votes
2answers
89 views

Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?

Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? More generally, for any Salem number $s$, ...
1
vote
0answers
34 views

Inertia of an elliptic curve with potentially good reduction

Let $E/\mathbb{Q}_p$, $p\geq5$ be an elliptic curve with additive potentially good reduction. Then there is a unique, minimal, finite and totally ramified extension $K$ such that $E/K$ has good ...
4
votes
2answers
68 views

Galois group of $\overline{\mathbb{F}_{p}}$ gives arithmetical information for finite fields $K/\mathbb{F}_{p}$?

Let $\mathbb{F}_{p}$ be the field with $p$ elements and $\overline{\mathbb{F}_{p}}$ be its algebraic closure. For some reason, we want to understand the structure of the Galois group of such an ...
2
votes
2answers
43 views

Finding the degree of $\sqrt[3]{2} + \sqrt[3]{3}$ over $\mathbb Q$ [duplicate]

I am practicing writing down random algebraic numbers and finding their degrees over $\mathbb Q$ and have fumbled when coming to $\sqrt[3]{2} + \sqrt[3]{3}$. Mathematica tells me the minimal ...
3
votes
0answers
37 views

Image of the norm map in imaginary quadratic fields

Let $K=\mathbb{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$. I want to know the image of the norm map $$ N^K_{\mathbb{Q}}:\mathcal{O}_K\to\mathbb{Z} $$ and the values of $N^...