Questions related to the algebraic structure of algebraic integers

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31
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342 views

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
3
votes
1answer
63 views

confusion with calculating the ideal class group of a quadratic field

I am a bit confused with the procedure of calculating the ideal class group of a quadratic field. From what I understood the computation starts by finding the Minkowski's bound say $n$. Then we list ...
1
vote
1answer
52 views

Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.

I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h). Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also ...
3
votes
0answers
23 views

Rational prime with specified factorization in $\mathbf{Z}[\mu_q]$

Let $r$ and $f$ be given positive integers. Prove that there exist primes $p$ and $q$ such that $p\mathbf{Z}[\mu_q]$ (where $\mu_q$ is a primitive $q$th root of unity) is a product of exactly $r$ ...
4
votes
1answer
98 views

When is $\mathbb{Z} [x]/f(x) $ a Dedekind domain?

Given a monic separable irreducible polynomial $f$ with integer coefficients, when $\mathbb{Z} [x]/f(x)$ is a Dedekind domain? And when it happens to be a Dedekind domain, how to know its class ...
2
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0answers
63 views

Evaluating the norm of an ideal by considering integral basis for $\mathcal{O}_K$

There is this trick in my lecture notes that I don't think is correct. Let $K=\mathbb{Q}(\sqrt{-5})$ then my notes say that the ideal $(2,1+\sqrt{-5})$ in $\mathcal{O}_K$ has obviously norm $2$ since $...
1
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1answer
31 views

Results in algebraic number theory regarding ramified split and inert primes in quadratic fields

I am currently reading some notes in algebraic number theory but they are not really self contained and I am guessing the following results must hold. Let $K$ be a quadratic field and consider the ...
0
votes
0answers
57 views

Intuition for Adeles and Ideles

I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? ...
3
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0answers
36 views

possible norms of prime ideals in the class group of $K=\mathbb{Q}(\sqrt{-21})$

I have an example in my notes where we try to compute the class group of the quadratic field $K=\mathbb{Q}(\sqrt{-21})$. My notes then proceed to evaluate the Minkowsk's bound< $\lambda(\sqrt{-21})$...
1
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0answers
85 views

Prime ideals of the ring of integers lying over $p\mathbb{Z}$ [duplicate]

Let $A$ be the ring of all elements of $\mathbb{C}$ that are integral over $\mathbb{Z}$, and $p\in\mathbb{Z}$ a prime element. Are there infinitely many prime ideals of $A$ lying over $p\mathbb{Z}$? ...
1
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1answer
42 views

Why assuming that the ideal in Minkowski's bound is prime

Minkowski’s bound states that given a quadratic field $K(\sqrt{d})$ then every class of ideals in $\mathcal{O}_K$ contains an integral ideal of norm<$\lambda(d)$. Then my notes say that this ...
0
votes
1answer
16 views

Definition of split prime in quadratic fields

I have the following definition of split prime number $p \in \mathbb{Z}$ in my lecture notes that I don't understand. Let $K$ be a quadratic field, the definition I have says: $p$ is called split in $...
3
votes
2answers
81 views

The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
0
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0answers
28 views

Prime Decomposition of an ideal in a number field.

I have been stuck on the 26th problem of the 3rd chapter from Marcus' Number Fields. Let $\alpha=\sqrt[3]{m}$ where $m$ is a cubefree integer, $K=\mathbb{Q}[\alpha]$, $R=\mathbb{A} \cap \mathbb{Q}[\...
0
votes
1answer
46 views

Ring of algebraic integers

Let $R$ be a ring of algebraic integers, $p ∈\mathbb{Z}$ a prime integer. Then the set $\mathcal A$ of all prime ideals $P ⊂ R$ such that $P ∩ \mathbb{Z}= p\mathbb{Z}$ is finite and nonempty. Also, $...
0
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0answers
54 views

A Problem from Marcus' Number Fields

I have been stuck on the 17th problem of the 3rd chapter from Marcus' Number Fields. Let $K=\mathbb{Q}[\sqrt-23]$ , $L=\mathbb{Q}[\omega]$ where $\omega = e^{2.\pi.i/23} $ . Let $P$ be one of the ...
1
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0answers
16 views

Maximal $\mathbb Z_p$-orders in $\mathbb Q_p[G]$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. ...
1
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0answers
33 views

The tower of ramification indices

Let $K\subset L\subset M$ be an extension of number fields. Let $R\subset S\subset T$, be their algebraic integers rings, respectively. Suppose that $P\subset R$, $Q\subset S$, $U\subset T$ be a prime ...
1
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0answers
30 views

Geometric Proof that Gaussian Integers form PID

In my reading on Algebraic Number Theory, I came across a description of a proof that the Gaussian Integers form a PID- you select a nonzero element $\alpha$ of minimal norm in the ideal, where the ...
2
votes
0answers
33 views

Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
0
votes
0answers
21 views

Definition of $S$-ideles

This is a basic notational question. Let $K$ be a number field and $M_K$ the set of all places of $K$ with $S\subset M_K$ a finite subset. Write $\mathfrak J_K$ for the idele group of $K$ and $\...
2
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0answers
29 views

Arithmetic in a dihedral extension

Let $\;$L = $\Bbb Q$[$\sqrt[4]{2}$, i ]$\;$ which is a dihedral extension of the rationals. There are three quadratic and five quartic intermediate fields between L and $\Bbb Q$. The following ...
2
votes
1answer
34 views

Basis of a Cyclotomic Field

I've started learning algebraic number theory when I found something that confused me; for a prime $p$, where $\zeta=e^{(2\pi i/p)}$, a primitive $p$-th root of unity. Then the extension $\mathbb{Q}[\...
5
votes
1answer
70 views

Exists sequence converging to $0$ in $\mathbb{R}$, $1$ in $\mathbb{Q}_2$?

Does there exist a sequence of elements $x_1, x_2, x_3, \ldots$ of elements of $\mathbb{Q}$ that converges to $0$ in $\mathbb{R}$ and converges to $1$ in $\mathbb{Q}_2$?
0
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0answers
29 views

Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} \chi(c_0)^p+\chi(...
1
vote
2answers
68 views

Galois group is isomorphic to group of characters. What can we say then?

Let $\overline{K}/K$ denote the separable closure of a finite field $K$ of characteristic $p$ and let $\mu_{n}$ denote the group of $n$-th roots of unity in $\overline{K}$, where $(n,p)=1$. Let us ...
0
votes
0answers
18 views

An ideal of $\mathcal O_K$ [duplicate]

Let $K$ be a number field and let $\mathcal O_K$ be its ring of integers. Let $I$ be a non-zero ideal in $\mathcal O_K$. If $I$ is free as a $\mathcal O_K$-module then is it a principal ideal? ...
7
votes
0answers
69 views

If $K\cap\Bbb Q^{\text{cycl}}=\Bbb Q(\zeta_m)$ and $K/\Bbb Q$ Galois, then $\text{Gal}(K(\zeta_n)/K)\cong\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q(\zeta_m))$

$\DeclareMathOperator{\Gal}{Gal}$ Here is my argument: Induction on the number of primes dividing $n/m$. If there are two primes (i.e., $K(\zeta_n) = K(\zeta_{q_1},\zeta_{q_2})$, where $q_i=p_i^{e_i}$...
1
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0answers
19 views

Double cosets (Neukirch's Algebraic Number Theory)

This is a question from Neukirch's Algebraic Number Theory, Ch.1 $\S$9. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable field extension of $K$ and $B$ the integral ...
3
votes
2answers
71 views

Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
0
votes
3answers
100 views

Showing reducibility of a polynomial in a Discrete Valuation Ring

Let $R$ be a complete discrete valuation ring with uniformiser $\pi$. I would like to show that a polynomial $f$ in $R[X]$ is reducible. Does it suffice to show that $f$ is reducible in $\frac{R}{\pi^...
2
votes
2answers
33 views

ramification index in an example

Let $L=\mathbb{Q}_5[x]/(x^4+5x^2+5)$, where $\mathbb{Q}_5$ is the field of 5-adic numbers. Note that the polynomial that we are quotienting out by is an Eisenstein polynomial. So $L/\mathbb{Q}_5$ is a ...
0
votes
1answer
26 views

Neukirch Abstract Kummer Theory. Understanding a Proof.

This question is a sort of follow up to this question, where I introduced context. Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT) The thing I don't understand know is ...
0
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0answers
36 views

$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
2
votes
1answer
54 views

Algebraic number theory, Marcus, Chapter 3, Question 9

Question 9 in Marcus book. Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively. a) Let $I$ and $J$ be ideals in $R$, and suppose $IS|...
1
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3answers
45 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
0
votes
1answer
53 views

Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT)

Someone familiar with Neukirch's terminology can understand this post better. Unfortunately it is so much terminology to just explain it here. My question is about what is marked in the picture: Why ...
1
vote
1answer
24 views

Algebraic number with conjugates having modulus 1

Suppose $\alpha$ is an algebraic number lying in a number field $K$ that is a normal extension of $\mathbb{Q}$. Suppose all the conjugates of $\alpha$ have absolute value 1. Prove or disprove that it ...
3
votes
1answer
57 views

Computing the Picard group of Number Fields in analogy with genus $0$ curves

It is possible to compute the Picard group of (some? all?) genus $0$ curves in the following manner: For concreteness let, $A = k[x,y]/(x^2+y^2-1)$ and $X = \operatorname{Spec} A$. Let $Y$ denote $\...
6
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1answer
119 views

Minimal polynomial of root of unity over quadratic field

Let $p$ be an odd prime and consider the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$ and its quadratic subfield $\mathbb{Q}(\sqrt{\pm p})=:K$. I am interested in the minimal polynomial of a root of ...
2
votes
1answer
98 views

Semilinear root of uniformiser of a p-adic field (& phi-module of Lubin–Tate formal group)

I'm looking for solutions $t$ of an equation of the form $$ t \sigma(t) \cdots \sigma^{n-1}(t) = v $$ in a field equipped with an automorphism $\sigma$ of order $n$. In this case, I call $t$ a "$\...
3
votes
0answers
60 views

Enumeration of the number of splitting fields

Suppose $f(x):=x^p+ax+b\in \mathbb Z[x]$ and let $S_f$ be the minimal splitting field of $f(x)$. How can we estimate $\#\{(a,b):|S_f:\mathbb Q|=2p\}$?
3
votes
0answers
59 views

“Logarithmic derivative of a p-adic number”

Coleman, in "Division Values in Local Fields", (Inventiones math. 53, 91 - 116 (1979)), says that In his work on cyclotomic fields Kummer observed that various formal operations on power series ...
15
votes
2answers
265 views

Primes of form $a^2 + 24b^2$

For a prime number $p \neq 2$, $3$, is it necessarily the case the prime number can be written in the form $a^2 + 24b^2$ if and only if $p \equiv 1 \text{ mod }24$? I think this has to be true based ...
1
vote
1answer
61 views

Absolute Galois group of quadratric extensions

It is well known that the absolute Galois group of $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{q})$ are nonisomorphic if $p$ and $q$ are different prime numbers. See for example Szamuely's book "...
1
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0answers
21 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
2
votes
1answer
43 views

Neukirch's Abstract CFT. Help with a proof in abstract Kummer theory.

First of all, unfortunately, writing all the notation and terminology that he uses would make this post very big. So, I'm really hoping from an answer that comes from someone that knows this book. ...
2
votes
1answer
55 views

Algebraic Number Theory,Marcus, Chapter 2, Question 16

In question 16 of chapter 2 in Marcus Book, I have to show that $\sqrt{3}\not\in\mathbb{Q}(\alpha)$,where $\alpha=\sqrt[4]{2}$ using the trace idea. the proof starts by assuming that $\sqrt{3}=a+b\...
3
votes
1answer
76 views

$p = x^2 + xy + 3y^2$ if and only if $p \equiv 1$, $3$, $4$, $5$, $9$ mod $11$? [duplicate]

For a prime number $p \neq 11$, do we have $p = x^2 + xy + 3y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1, 3, 4, 5, 9$ mod $11$? An example where this is true:$$5 = 1^2 + 1 \times ...
1
vote
0answers
30 views

Roots of unity in fixed field of decomposition group.

$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next? Suppose ...