Questions related to the algebraic structure of algebraic integers

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2
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1answer
65 views

Why $ \mathbb{Z}[x]$ is not Principal Ideal Domain [duplicate]

$ \mathbf{Z}[x]$ is not PID. we know $\mathbb Z$ is a Unique Factorization Domain, so $\mathbb Z[x]$ is UFD, but why isn't it PID (since I think $\mathbb Z$ is PID)?
3
votes
0answers
31 views

Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that ...
0
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0answers
29 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
7
votes
2answers
75 views

Units of $\overline{\mathbb{Z}}$

What are the units of $ \overline{\mathbb{Z}} $ (the ring of algebraic integers)? I know all roots of monic polynomials with constant term 1 are units, but are there any others?
3
votes
2answers
62 views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...
0
votes
1answer
45 views

Compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$

How do I compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$? For example, $\alpha = \sqrt[3]{-19}$ and $\beta = (\alpha^2 - \alpha + 1)/3$ satisfy $(\alpha + 1)\beta = ...
7
votes
0answers
60 views

If $p\equiv 1,9 \pmod{20}$ is a prime number, then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$.

I have to prove that if $p\equiv 1,9 \pmod{20}$ is a prime number then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$. I consider the quadratic field $\mathbb{Q}(\sqrt{-5})$, with ring of ...
1
vote
0answers
17 views

Measurability in the proof of Minkowski's Bound for calculating the Class Number?

I recently looked at the proof of Minkowski's Bound given in Number Rings, and it appeared to implicitly rely on the interesting (and somewhat non-trivial) fact that convex subsets of $\mathbb{R}^n$ ...
2
votes
1answer
39 views

Inertia groups of cyclotomic field extension

I would like to ask about the inertia group of primes in the cyclotomic field extension. Indeed, let $n=p_{1}\cdots p_{k}$ for some distinct odd primes and let $K = \mathbb{Q}(\zeta_{n})$ so the ...
4
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1answer
30 views

Picard group of $\mathbb Z[\sqrt{-5}]$

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of ...
2
votes
1answer
62 views

Describing $Spec(\mathcal{O}_K[X])$

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. I am trying to describe $Spec(\mathcal{O}_K[X])$ in terms of fibers of the map $g: Spec(\mathcal{O}_K[X]) \rightarrow ...
2
votes
1answer
38 views

Is $\mathcal{O}_K$ always isomorphic to $\mathbb{Z}[X]/(f(x))$, for some irreducible polynomial $f(x)$?

Given an algebraic number field $K$ and its ring of integers $\mathcal{O}_K$, is $\mathcal{O}_K$ always isomorphic to $\mathbb{Z}[X]/(f(x))$, for some irreducible polynomial $f(x)$? Since ...
0
votes
0answers
26 views

Showing that $A^*$ is and ideal and that “$*$” is multiplicative.

Let $E/F$ be an extension of algebraic number fields and $\mathcal{O}_E$ and $\mathcal{O}_F$ be the ring of integers. Define $$A^*=\mathcal{O}_EA,$$ where A is an ideal of $\mathcal{O}_F$. Prove that ...
1
vote
1answer
26 views

Convergence of roots of polynomials with coefficients in non-archimedean local field

Let $K$ be a local field with non-archimedean absolute value $|\cdot|$ and ring of integers $\mathcal{O}_K := \{x \in K : |x| \leq 1\}$. Moreover, let $n$ be a positive integer. It can be proved that ...
6
votes
2answers
82 views

Infinite Units for $\mathbb{Z}[\sqrt{7}]$

Suppose that $\alpha \in \mathbb{Z}[\sqrt{7}]$ where $\alpha$ is of the form $a + b\sqrt{7}$ where $a, b \in \mathbb{Z}$. Because $\alpha$ is a unit if and only if $N(\alpha)=\pm 1$ we must show: ...
0
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0answers
25 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
2
votes
1answer
26 views

Aside from $\langle 0 \rangle$, can a ring of algebraic integers have prime ideals that are not maximal?

I have a feeling that a ring with such ideals would have to be non-UFD, and I can prove that in $\mathbb{Z}$ there are no such ideals. But in other rings, I'm not so sure. I'm not yet at a point at ...
0
votes
0answers
13 views

Set of squares in quadratic forms of a given discriminant.

For quadratic forms of negative discriminant, the set of squares is the same as the principal genus $H$ (forms whose values in $Z/DZ$ is the same as that of $x^2 + ny^2$ or $x^2 + xy + ny^2$ where ...
1
vote
1answer
23 views

Product involving Dirichlet characters $\prod_{i=1}^{\phi(m)}(1 - \frac{\chi_i(p)}{p^s}) =(1 - \frac{1}{p^{fs}})^{\frac{\phi(m)}{f}}$

While working with divisors in cyclotomic extensions of $\mathbb{Q},$ I came across this identity: Given a prime $p$ and an integer $m$ such that $(p,m) =1$, let $f$ be the smallest such the $p^f ...
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0answers
24 views

Algebraic integers in $p^r$-th cyclotomic field

I am having some troubles in understanding the proof of Theorem 10, Chapter 2 of D. Marcus' book Number fields, which shows that the ring of algebraic integers in the $p^r$-th cyclotomic field ...
2
votes
1answer
28 views

Do these properties characterize number rings?

Suppose $R$ is a Dedekind domain with the following properties: at every prime of $R$ the residue field is finite; fibers of the map $\text{Spec }R\to \text{Spec } \mathbb Z$ are finite. Is $R$ ...
3
votes
1answer
28 views

Describing integral closure of quadratic number fields

I'm facing the following problem. Let $p$ be a prime and $ K=\mathbb{Q}(\sqrt{p}) $. I'm trying to find the integral closure of $ \mathbb{Z} $ in $ K $. I don't really know where to start. I've ...
0
votes
1answer
14 views

For $m$ distinct fields among $\mathbb{Q}(\theta_1),\ldots,\mathbb{Q}(\theta_n)$ prove that $m\mid n$ and each field occurs $n/m$ times

I'm having some trouble with this problem, and I wanted to know if someone could help me out. Let $K=\mathbb{Q}(\theta)$ be an algebraic number field of degree $n$. Let ...
0
votes
1answer
32 views

Good book for Local Fields/ Commutative algebra?

I am currently studying Local Fields from Serre's textbook, but finding that it requires a bit too much prior knowledge for me. Can anyone suggest another book that I can use alongside Serre that ...
0
votes
1answer
25 views

How to construct a polynomial from a radix-term?

A term only composed of the following operatings shall henceforth be called a radix term because I don't know how these terms are called. A radix term $t$ is either an integer or a sum of two radix ...
0
votes
1answer
29 views

Understanding the proof (via primary decomposition) the “ideal factorization theorem” in Dedekind domains

I am trying to understand the outline of the strategy for proving (via primary decomposition) that every non-zero ideal of a Dedekind domain can be expressed uniquely (up to the order of the factors) ...
1
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0answers
24 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
2
votes
0answers
22 views

Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
2
votes
0answers
55 views

Discriminant of a field extension

I'm struggling with this exercise: Let $K:=\mathbb{Q}(\sqrt{2},\sqrt{-1})$ and $R:=\mathbb{Z}(\sqrt{2},\sqrt{-1})$. a) Compute the discriminant of $R$ over $\mathbb{Z}$. Suppose that $S$ is ...
3
votes
1answer
79 views

Dirichlet density

How to solve the following exercise: Let $q$ be prime. Show that the set of primes p for which $p \equiv 1\pmod q$ and $$2^{(p-1)/q} \equiv 1 \pmod p$$ has Dirichlet density ...
2
votes
0answers
33 views

Question on Dirichlet density

I did not understand the highlighted sentence of the exercise below: My question is: how does it follow that $f(x)=0$ has a solution mod $p$ implies that $f(x)$ (mod $p$) splits as the product of ...
1
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0answers
41 views

Definition of module

In the book's, The theory of numbers, S. Iyanaga. Chater I, Cohomology of groups. What is the meaning of "module A"? Thank you all.
2
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4answers
64 views

Counterexample in Dedekind domains

Let $K$ be a number field and $\mathcal O_K$ the ring of algebraic integers in $K$. If $\mathfrak p$ is a prime ideal, then $\mathcal O_K/\mathfrak p$ is finite field. My question is: Finding a ...
0
votes
0answers
22 views

Bounding height of an algebraic number

Let $\zeta_l$ be a primitive $l^{th}$ root of unity($l\geq 5$) and let $H$ be the height of an algebraic number(see for instance page 230 of The Arithmetic of Elliptic Curves by J. Silverman). I ...
1
vote
1answer
38 views

References for Algebraic number theory

I am doing algebraic number theory first time. I have done all ring theory and field theory. I am interested in algebra , so also pretty much excited about algebraic number theory. I have a month's ...
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0answers
18 views

Specific question on dirichlet density

In a notes I found the following exercise and solution: I have a question. In the proof I admit the statement "the Dirichlet density of these prime ideals is $1/2$ " but i do not understand why the ...
0
votes
0answers
15 views

Representation groups over Dedekind domains

I am interested on groups defined over $O_K$ the ring of integers of a number field $K$. Given a linear representation $T:Gl_N(O_K)\rightarrow Gl(W)$ with $W$ a free $O_K$-module, What are the main ...
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0answers
48 views

Isomorphism for the group of units of the ring of integers of a local field

Let $K$ be a local field with a discrete and non-archimedean absolute value, $\mathcal{O}_K$ be its ring of integers, $\mathfrak{m}_K$ be the unique maximal ideal of $\mathcal{O}_K$ and ...
2
votes
1answer
67 views

Galois theory on curves

Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line ...
2
votes
0answers
27 views

Find out how the prime numbers 2, 3, 37 splits in K = Q(√37) [closed]

Find out how the prime numbers $2,3,37$ split in $K=\Bbb Q(\sqrt{37})$; i.e. find $r$ and thos $e_i,f_i$ for $1\leq i\leq r$. How do I solve this. Thanks!
4
votes
2answers
63 views

Given that there is at least one solution to $a^{2} + 2b^{2} = p^{11}q^{13}$, find how many integers solutions there are.

I cannot even begin this problem, given $ a, b \in \mathbb{Z}$ and $p,q$ odd prime numbers, given that there is a soltuion to the equation: $a^{2} + 2b^{2} = p^{11}q^{13}$, find how many solutions ...
3
votes
2answers
80 views

Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
2
votes
1answer
38 views

Special case of Kronecker–Weber theorem.

Let $K$ be a number field contained in $m^{th}$ cyclotomic field, that is $K \subset \Bbb{Q}(\omega)$ where $\omega$ is a primitive $m^{th}$ root of unity. Let $p^k$ be the exact power of a prime $p$ ...
0
votes
1answer
23 views

Nontrivial characters of $(\Bbb{Z}_m)^{\ast}$

I was reading the book of Marcus on Number field page 196. I could not understang the highlighted equality. It will be helpful if someone gives me a proof. Thanks in advance for the computation!
3
votes
0answers
63 views

How to calculate the ideal class group of a quadratic number field?

The books I use to study Algebraic Number Theory are rather thin on the ground with concrete examples, so I make my own and check the results with Sage. To get some more hands on experience I want ...
2
votes
0answers
19 views

Primes Representable as Quadratic Form - Specifically Norm of an Imaginary Quadratic Field with Class # 1

Could someone point me towards the result that states that a prime is expressible as a norm of an imaginary quadratic field with class number 1 iff $\left(\frac{p}{D}\right)=1$.
8
votes
2answers
231 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
2
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0answers
22 views

another representation of the zeta function of a curve over a finite field

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
4
votes
1answer
26 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
0
votes
0answers
20 views

Is it true that $B=[\beta]$ when $B$ is and ideal of $\mathcal{O}$, $\beta \in B$ and $N(B)=|N(\beta)|$?

Let $B$ be an ideal of $\mathcal{O}$ (Ring of integers), $\beta \in B$ and $N(B)=|N(\beta)|$. Does it follow that $B=[\beta]$? I think that this isn't true but I'm struggling to find a counter ...