Questions related to the algebraic structure of algebraic integers

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6
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0answers
95 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*} ...
1
vote
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
votes
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 ...
-2
votes
2answers
84 views

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

So far I've got something like $-\frac{a\sqrt[3]{n}}{2} \pm \frac{b\sqrt{-3}}{2c}$.
0
votes
0answers
27 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
71 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
276 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
34 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
37 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
votes
1answer
31 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 ...
0
votes
2answers
33 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
60 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
38 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
votes
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
2
votes
2answers
58 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
102 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 ...
1
vote
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
53 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
87 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
38 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
181 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
37 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
54 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
91 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
97 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 ...
4
votes
2answers
55 views

density theorems and class field theory

am I correct in thinking that the frobenius density theorem (it says that the Dirichlet density of the set of primes of K that split completely in an extension L is 1/[L:k]) is sort of one of the main ...
1
vote
1answer
28 views

$ K(\sqrt{a})$ is unramified if and only if $a \mid d_K$ and $a \equiv 1 \mod{4}$.

Let $K$ be an imaginary quadratic field of discriminant $d_K$ and let $K(\sqrt{a})$ be a quadratic extension where $a \in \mathbb{Z}$. Then $K \subset K(\sqrt{a})$ is unramified if and only if $a$ can ...
4
votes
1answer
151 views

Where does a CM elliptic curve have bad reduction?

Let $d>1$ be square-free, and $K=\mathbf Q(\sqrt{-d})$. Choose an embedding of $K$ in $\mathbf C$, and let $E = \mathbf C/\mathcal O_K$. It is known that $E$ admits a model over the Hilbert class ...
1
vote
1answer
28 views

Divisibility in the ring of integers.

For example, let $R=\Bbb Z [\sqrt{-5}]$, and I want to explain $3$ does not divide $2-\sqrt{-5}$. I think the following proof will be right: Suppose $3(a+b\sqrt{-5})=2-\sqrt{-5}$, then taking ...
1
vote
0answers
30 views

Is there any concept similar to unique factorization that applies to exponential operators?

We can talk about prime numbers over multiplication but is there any similar concept that applies to exponential operators or other hyperpowers like tetration? Can we use what we know about UFDs to ...
4
votes
0answers
46 views

Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
3
votes
1answer
54 views

Minkowski's theorem for non-symmetric convex bodies

Minkowski's theorem for convex bodies states that every convex, symmetric subset of $\mathbb{R}^d$ whose volume is larger than $2^d$ contains a non-zero integer point. All the proofs I've seen rely on ...
1
vote
1answer
47 views

Degree of extension is equal to linear combination of prime factor multiplicities with prime factor index coefficients in Dedekind domains

I'm working on the following problem... Suppose that $A$ is a Dedekind domain with fraction field $K$. $L/K$ is a finite separable extension of $A$ of degree $n$, and $B$ is the integral closure ...
1
vote
1answer
45 views

Are there number fields based on ultraradicals?

So you can create quadratic fields, cubic fields, and quartic fields by just taking the nth root of some integer, and some are even unique factorization domains or principle ideal domains, like $\Bbb ...
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vote
0answers
14 views

Are there unique factorizations for weyl algebras?

I just read about Weyl algebras, and they sound like neat little toys that are similar in a number of ways to polynomials. However its curious to me that they are non-commutative, and I was wondering ...
1
vote
4answers
56 views

Number theory - rational number

Are there any $x, y$ that fit in below $\sqrt{4y^2-3x^2}$ such that an rational number is yielded. Appreciate if explanation is given.
1
vote
1answer
53 views

Prime number of ${\bf Z}$ and prime element of ${\bf Z}[i]$

I am looking at the class note from graduate number theory: Let $p$ be prime number in ${\bf Z}$ and r be prime element in ${\bf Z}[i]$. If $r$ is an associate of $p$, then $p$ is congruent to $3$ ...
0
votes
0answers
24 views

Methods for computing subextensions for a n-th cyclotomic field.

So the problem is 1)find all quadratic and cubic subextensions of $\mathbb{Q}[\zeta^{527}]$ and 2)describe how it's primes split completely in the cubic subextensions. Can you give me some ...
1
vote
1answer
54 views

Is $f(x)=x^{4}-2x^2 +3$ Eiseinstein in 2-adic $\mathbb{Q}_{2}$?

I think it is because $|1|_{2}=1$, $|2|_{2}=2^{-1}\leq 1$ and $|3|$,where 3 is prime I am using the following Eisenstein criterion for $f(x)=a_{n}x^{n}+...+a_{0}$: $|a_{n}|=1$, $|a_{0}|=prime$ and ...
0
votes
0answers
16 views

Discriminant of p-adic $\mathbb{Q}_{p}[\phi]$, where $0=f(\phi)=\phi^{p}-\phi-1$

Any suggestions using the minimal polynomial? How about $D_{\mathbb{Q}_{p}[\phi]/\mathbb{Q}_{p}}=(-1)N_{\mathbb{Q}_{p}[\phi],\mathbb{Q}_{p}}(f')$? But foremost I prefer you suggest me a correct ...
0
votes
2answers
46 views

Finding number of roots of a polynomial in p-adic integers $\mathbb{Z}_{p}$

The problem is to find the number roots of $x^3+25x^2+x-9 $ in $\mathbb{Z}_{p}$ for p=2,3,5,7. I read this equivalent to having a root mod $p^{k}~\forall k\geq 1$. By Newton's lemma I can get whether ...
0
votes
1answer
70 views

$(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$ if and only if $n$ is not a square mod $p$

Let $n$ be a square-free integer such that $n\equiv 2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
0
votes
1answer
23 views

Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...