Questions related to the algebraic structure of algebraic integers

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3
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what is the unique prime factorization for the ideal $p\mathbb{Z}[\zeta]$ in the Dedekind domain $\mathbb{Z}[\zeta]$?

Let $p$ be a prime number and $\zeta=e^{\frac{2\pi}{p}i}$. I want to find the unique factorization into a product of prime $\mathbb{Z}[\zeta]$-ideals for the ideal $p\mathbb{Z}[\zeta]$. Now, I know ...
1
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0answers
55 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
2
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0answers
33 views

$\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$

prove that $\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$ my aim is to show that $\Bbb{Q}_K=\Bbb{Z}+\sqrt[3]{3}\Bbb{Z}+(\sqrt[3]{3})^2\Bbb{Z}$ $\supseteq$ : ...
3
votes
2answers
60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
2
votes
2answers
125 views

Silly mistake in this number theory book

My question is very easily to be solved (at least I hope so) I think this book has a mistake: When I calculate I get $b_3\equiv -2 (\mod{2})$ which implies $b_3=0$, am I right? Another question, ...
0
votes
1answer
57 views

Is this ring an integral domain?

I'm starting to study Algebraic number theory and I'm having problems with the first examples of this book. I'm trying to prove this is a quadratic domain, i.e., an integral domain: I'm sorry I ...
1
vote
1answer
53 views

Integral Closure in $\mathbb{Q}(\alpha)$

Let $\alpha $ be a root of $f(x) = x^3 -x +3$, $K=\mathbb{Q}(\alpha)$ I have to prove $\mathcal{O} _K = \mathbb{Z}[\alpha]$ for the integral closure $\mathcal{O} _K$. I have shown that the disciminant ...
1
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0answers
22 views

$\zeta$ primitive $n$th root of unity, help showing that $\sqrt{n},\sqrt{-n}\in \mathbb{Q}(\zeta)$ under some conditions.

Consider $\zeta$ a primitive $n$th root of unity, show that if $n\equiv 1\mod{4}\implies \sqrt{n}\in \mathbb{Q}(\zeta)$ if $n\equiv -1\mod{4}\implies \sqrt{-n}\in \mathbb{Q}(\zeta)$. I know that ...
2
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0answers
31 views

Dedekind's criterion clarification

Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf) Is it true that the prime ideals ...
12
votes
1answer
124 views

When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that ...
1
vote
1answer
24 views

Normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ is UFD…

We know that the normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ where $d\in \mathbb{Z}$ is $$O=\mathbb{Z}[\beta], \text{$\beta=\sqrt{d}$ if $d\equiv2,3 \pmod 4$}; \ \frac{1+\sqrt{d}}{2} ...
6
votes
1answer
75 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
5
votes
2answers
75 views

Subgroup of class group

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, ring of integers $\mathcal O_K$ and $\mathcal Cl(K)$ its ideals class group. I want to show that $\mathcal Cl(K)^G$ is generated ...
6
votes
1answer
122 views

What is the largest possible length of a prime number?

Let $p$ be a prime number , set $f(p)=2p+1$ and define $f^n(p)=f\circ f\circ\cdots\circ f(p)$ composition by $f,$ $n$ times. And define length of $p$, $L(p)$ as maximum of $n$ such that $f^i(p)$ is ...
3
votes
1answer
33 views

Ramification of the prime at infinity of function fields

Let $K=\mathbb{F}_q(x)$ be a function field and $L = \mathbb{F}_q(X)(\sqrt[n]{f(x)})$ be an extension where $\deg(f) = d$. I want to find what the ramification index at infinity is. If we let $s = ...
3
votes
1answer
62 views

On Hilbert Class Polynomial

Is there any open source software which computes Hilbert Class polynomial of an imaginary quadratic fields? Thank you in advance
31
votes
1answer
597 views

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 $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
6
votes
0answers
71 views

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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2answers
33 views

Is $\Bbb{R}$ finitely generated over the algebraics?

Let $K$ be the field of all algebraic numbers. Then does there exist an algebraically independent set of numbers $\pi_1, \dots, \pi_n$ such that $\Bbb{R} = K(\pi_1, \dots, \pi_n)$? I don't know ...
2
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0answers
38 views

The number of abolute value on $\mathbb{Q}(\sqrt{2})$

Let $|.|$ be the usual absolute value on $\mathbb{Q}$. The number of absolute value on $Q(\sqrt{2})$ extending |.| is 2 since $x^2-2=(x-\sqrt{2})(x+\sqrt{2})$ in $\mathbb{R}[x]$. Let ...
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votes
2answers
83 views

What's the formula for the complex cubic roots of an arbitrary positive integer $n$? [closed]

So far I've got something like $-\frac{a\sqrt[3]{n}}{2} \pm \frac{b\sqrt{-3}}{2c}$.
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0answers
26 views

Factorization of prime ideals

Suppose $L/K$ is a finite extension of number fields, $I,J$ are ideals in $\mathcal{O}_K$ and $I\mathcal{O}_L|J\mathcal{O}_L$, prove that $I=I\mathcal{O}_L\cap \mathcal{O}_K.$ I knew that the ...
3
votes
2answers
69 views

Number of algebraic integer divisors of an algebraic integer

Let $\alpha$ be an algebraic integer of degree $d$. Let $\tau(\alpha)$ be the number algebraic integers $\beta$ of degree $d$ such that $\alpha/\beta \in \mathbb{Z}$. What is a good upper bound on ...
4
votes
3answers
274 views

Can the norm of a non-algebraic integer be an integer?

Let L/K be a finite field extension and define the norm of an element as the product of each K-embedding evaluated at that element. Can the norm of a non-algebraic integer be an integer? I know that ...
1
vote
0answers
42 views

More than two distinct factorizations?

So we all know about 6 and its two factorizations in $\mathbb{Z}[\sqrt{10}]$. Can an integer have more than two factorizations in some non-UFD? And if so, can it be a squarefree semiprime like 6 or ...
0
votes
1answer
33 views

Prove that $D(\alpha)=D(\beta)$

Let K be an algebraic number field. Let $\alpha \in$ K. Let $\beta$ be conjugate of $\alpha$ relative to K . Prove that $D(\alpha)=D(\beta)$. $D(\alpha)$:= Let K be algebraic number field of degree ...
0
votes
0answers
36 views

Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
0
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1answer
30 views

Free abelian group of fractional ideals

This question is from Ch.2 of Frohlich and Taylor's Algebraic Number Theory, page 42. Let $R$ be a Dedekind domain, $I_R$ the multiplicative group of fractional $R$-ideals. There is an isomorphism of ...
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2answers
32 views

Show that $x$ is an algebraic number? Where $x$ is…

Can someone help me with the following problem? Show that $x=\sqrt2+\sqrt[3]3$ is an algebraic number. By finding a polynomial with rational coefficients for which $x$ is a root of. Can someone ...
5
votes
0answers
58 views

Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio

I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...
4
votes
1answer
56 views

generalized ideal class group for infinitely many moduli (Cox 8.4)

I am given the following definition (without the proof or technical details). and I need to understand that I tried the following: Since $P_{K,1}(\mathfrak{m}) \subseteq ...
3
votes
2answers
55 views

$p$ ramifies in a number field, then it does so in an overfield

If $p$ ramifies in a number field $K$, and we have number field extensions $F:K:\mathbb{Q}$, does it follow that $p$ ramifies in $F$? Please give me some hints. If true, I'll need to work out a direct ...
2
votes
3answers
37 views

Find the product $(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$

Let $1,$ $a_i$ for $1 \leq i \leq 6$ be the different roots of $x^7-1$. Then find the product: $(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$ I don't know how to proceed.
0
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2answers
36 views

Prove that $x_n \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \Bbb Z_+$

Given that $x_n = \displaystyle \prod_{i=1}^n \frac{2i-1}{2i}$ Then prove that $x_n \leq \frac{1}{\sqrt{3n+1}}$ for all $n \in \mathbb Z_+$ What I did was take the logarithm of $x_n$, and I arrived ...
0
votes
1answer
60 views

Show that $n$ is prime. [closed]

Let $x$ and $n$ be positive integers such that $\displaystyle \sum_{i=0}^{n-1} x^i$ is a prime number Thus, show that $n$ is also prime
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0answers
24 views

Ireland and Rosen, question 12.28 (unique factorization in prime ideals)

I'm stuck with the following exercise in Ireland and Rosen, chapter 12. Let $D$ be the ring of integers in a number field $F$. Suppose $(p)=P^2A$ for $p$ prime in $\Bbb Z$ and a prime ideal $P$. ...
2
votes
2answers
56 views

Finding the Norm of an element in a field extension

If I have a field extension of $\mathbb{Q}$ given by $\mathbb{Q}(\alpha)$ and the only thing I know about the primitive element $\alpha$ is it's minimal polynomial $p(x) = a_0 + a_1x + ... + x^n$ such ...
2
votes
0answers
37 views

$\operatorname{Br}(\Bbb Q_{47})$

I'm looking for division algebras over $\Bbb Q_{47}$. I guess my best bet is to calculate the Brauer group $\operatorname{Br}(\Bbb Q_{47})$. What's the best way of performing this calculation? Should ...
0
votes
1answer
16 views

Norm of $(\alpha - a) = (-1)^{\deg f}f(a)$

Let $\Bbb Q(\alpha)$ be a number field, and $f$ the minimal polynomial of $\alpha$. Why is $N_{\Bbb Q(\alpha)/\Bbb Q} (\alpha-a)= (-1)^{\deg f}f(a)$? This works obviously for $a=0$ by the definition ...
4
votes
1answer
97 views

Localization in formal power series

I saw in a textbook the following assertion: Let $R$ be a commutative ring with unity, and $R[[X]]$ be the ring of power series in one indeterminate $X$. If the homomorphism $\phi∶ R[[X]] \to R$ ...
5
votes
2answers
41 views

Irreducibility of polynomials of the form $x^p - n$ over the cyclotomic field $Q(\zeta_p)$?

Is there a general procedure for showing that the polynomial $x^p - n$ is irreducible over the cyclotomic field $Q(\zeta_p)$? ($\zeta_p$ a primitive pth root of unity, and $n \in \mathbb{N}$. Maybe ...
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0answers
22 views

All the isomorphisms of a finite algebraic separable field extension

I'm new to algebraic number theory and field extension theory. From what I've understood, a finite algebraic field extension $L/K$ is a vector space over $K$ of dimension $n$ and can be seen as ...
2
votes
1answer
50 views

Computing $H^\bullet(\Bbb Z/n\Bbb Z)$

This is related to this other question of mine Showing that $\operatorname {Br}(\Bbb F_q)=0$ in which I also got stuck at writing a free resolution. I want to compute the group cohomology ...
5
votes
2answers
82 views

Showing that $\operatorname {Br}(\Bbb F_q)=0$

I want to prove that $\operatorname {Br}(\Bbb F_q)=0$ using the cohomological description of the Brauer group. We have: $\operatorname {Br}(\Bbb F_q)=H^2(\operatorname {Gal}(\overline {\Bbb ...
2
votes
1answer
36 views

$\Bbb Q (\sqrt{-535}, \sqrt 5)$ is unramified over $\Bbb Q (\sqrt {-535})$

From the calculation of the discriminant, I know that the extension $\Bbb Q (\sqrt {-535})/\Bbb Q$ ramifies only at $2,5,107$. ($\Delta=4\cdot(-535)=-4\cdot5\cdot 107$) Since $\Bbb Q(\sqrt 5)/\Bbb Q$ ...
7
votes
2answers
178 views

Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
2
votes
1answer
35 views

Relationship between Ramification and Minimum Polynomial Factorisation

Consider the following set-up: Let $d \neq 0,1$ be a square-free integer and $p$ a prime. Let $K=\mathbb{Q}(\sqrt{d})$ and denote $\Delta^2=\Delta^2(K)$, the discriminant of $K$. I want to prove the ...
1
vote
1answer
50 views

What is known about the ramification index of ramified primes in an arbitrary cyclotomic extension of $\mathbb{Q}$

Let $\zeta$ be a primitive $m$th root of unity, and $L = \mathbb{Q}(\zeta)$. Then $B = \mathbb{Z}[\zeta]$ is the integral closure of $\mathbb{Z}$ in $L$. If $P$ is a prime ideal of $B$ and ...
1
vote
1answer
74 views

Minimal polynomials and degree of field extension

I have a cyclotomic field $\mathbb{Q}(\zeta_3)$, and want to know how I can find a minimal polynomial of $\zeta_{10}$, and $\zeta_{12}$. I have determined that both the polynomials should be of ...
3
votes
2answers
92 views

Cyclotomic polynomial in $\mathbb{Z}/(p)$

Let $p$ be prime, $n \in \mathbb{N}$ and $p \nmid n$. $\Phi_n$ is the $n$-th cyclotomic polynomial. How can I find the maximum $n \in \mathbb{N}$ (with $p \nmid n)$ so that $\Phi_n$ splits into ...