Questions related to the algebraic structure of algebraic integers

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1answer
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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
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 ...
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0answers
61 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
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1answer
58 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 ...
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2answers
59 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 ...
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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.
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2answers
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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 ...
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1answer
61 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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2answers
62 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 ...
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0answers
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$\operatorname{Br}(\Bbb Q_{47})$

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 ...
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1answer
17 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
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1answer
104 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
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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
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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
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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 ...
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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 $\Bbb Q(\sqrt 5)/\Bbb Q$ ...
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2answers
184 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 ...
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1answer
40 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 ...
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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 ...
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1answer
112 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 ...
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2answers
99 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
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2answers
56 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 ...
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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
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1answer
168 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
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1answer
31 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 ...
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0answers
33 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 ...
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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
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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 ...
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1answer
48 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 ...
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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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0answers
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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 ...
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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.
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1answer
54 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 $3$ ...
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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 ...
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1answer
55 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 ...
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0answers
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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 ...
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2answers
47 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 ...
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1answer
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$(p)$ 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 ...
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1answer
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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? ...
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2answers
71 views

Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
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Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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0answers
25 views

Integer Solutions of eliptic curve

I neet to find all integer solutions to the equation $2x^2+25=y^3$ Can you please help me?
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1answer
29 views

What does “A mod P generates the residue class field extension” mean?

We have K and finite algebraic extension L. P is a prime ideal in $O_{L}$ over prime $p\in O_{K}$ and $A\in O_{L}$. Then the problem says $\bar{A}:=A ~mod ~P$ generates the residue class field ...
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1answer
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Geometric meaning of being integrally closed in some overring

The geometric counterpart of integrally closed rings (in their fraction fields) are normal varieties, as described in this MathOverflow post. Is their a similar notion in algebraic geometry for being ...
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ANT Frohlich Proposition 3, (v). Induced map of dual modules has the same determinant

$R$ is a Dedekind domain, $V$ is an $n$ dimensional vector space over its quotient field $K$, $B(-,-)$ is a $K$-bilinear form on $V$, and $M, N \subseteq V$ are free $R$-modules of rank $n$. Also ...
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0answers
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Can the General Number Field Sieve be used to factor in any unique factorization domain?

Related slightly to my question about factoring in quadratic rings, can you use the general number field sieve to factor in any unique factorization domain? Can you use it in any UFD that isn't the ...
0
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1answer
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Uniqueness of unramified extensions of $\mathbb{Q}_{p}$

So I showed that $\mathbb{Q}_{p}[\theta]$ is an unramified extension of degree p, where $0=g(\theta)=\theta^{p}-\theta-1$. But it also follows that $\mathbb{Q}_{p}[\phi]$ is an unramified extension ...
3
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1answer
47 views

Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
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3answers
39 views

Ramification and inertia degree for $\mathbb{Q}_{p}[a]$ where $0=g(a)=a^{3}+25a^{2}+a-9$

The problem is to find e and f for p-adic rationals for p=2,3,5,7. Because g is not Eisenstein for each p, the extension will not be tottaly ramified and thus $3=ef\Rightarrow e=1$ and $f=3$. I feel I ...