Questions related to the algebraic structure of algebraic integers

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Square of an algebraic integer is an algebraic integer

For some $\alpha\in\mathbb C$ let $E=\mathbb Q(\alpha)$ be a number field and $\mathcal I$ its ring of integers. Suppose $(a_1+b_1\alpha)\cdots(a_k+b_k\alpha)=\beta^2$ for some $\beta\in E$ and ...
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2answers
89 views

Primes corresponding to embeddings of a number field

Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we ...
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1answer
221 views

Show that a number field is isomorphic to a quotient $\mathbb Q[x]/(f)$

Let $K$ be a number field of degree 3. Show that $K$ is isomorphic to a quotient $\mathbb Q[x]/(f)$, with $f = x^3 + ax + b$ in $\mathbb Z[x]$ irreducible in $\mathbb Q[x]$ (without using the result ...
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1answer
35 views

Free Fractional Modules

Let $R$ be a Dedekind domain with fractional ideal $M$. I am trying to prove: If $M$ is free, then the ideal $I$ such that $M=d^{-1}I$, where $d \in R$, is principal. Since $R$ is a Dedekind ...
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2answers
68 views

Why ideal containment in the proof of unique ideal factorization?

I want to understand the proof that the number fields (which are dedekind domains) have unique factorization of ideals. I am trying to read this proof here IDEAL FACTORIZATION - KEITH CONRAD but.. ...
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680 views

Ramified primes in a cyclotomic number field of a prime power order

Let $l$ be a prime number(even or odd), $n \geq 1$ an interger. Let $\zeta$ be a primitive $l^n$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Is the following proposition true? If ...
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119 views

$\hat{M_P}$ $\cong$ $\prod_{i}\hat{M_{Q_i}}$

I think I came up with the following result. But I'm not 100% sure. Is this correct? If yes, how does one prove this? Theorem? Let $A$ be a discrete valuation ring, $K$ its field of fractions. Let ...
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93 views

Modular forms have values as algebraic numbers?

Can you find examples of modular forms which take on values as e.g. algebraic numbers of degree n ? I'm interested in finding special classes of algebraic numbers particularly when n=3, they don't ...
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2answers
303 views

Ramified extension in number theory

Assume that $K$ is a complete field under a discrete valuation with Dedekind ring $A$ and maximal ideal $\mathfrak p$ and $A\diagup\mathfrak p$ is perfect. Let $e$ be a positive integer not divisible ...
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180 views

a calculation problem about haar measure

Here is a problem in charpter $2$ section $5$ in Algebraic Number Theory written by Jiirgen Neukirch. The problem is Let $K$ be a $p-adic$ number field, $v_p$ the normalized exponential ...
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70 views

What does Lang mean here by “the usual criterion”?

Let $K$ be a number field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones and all those which divide $n$, $K_S$ the group of $S$-units of $K$, and ...
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1answer
29 views

Prove that $L(\psi_N , s) = \zeta(s) \prod_{p \mid N}(1 − p^{-s})$

Let $N$ be a positive integer, and let $\psi_N$ be the trivial Dirichlet character with conductor $N,$ so $\psi_N (a) = 0$ if $\gcd(a, N) \ne 1$ and $\psi_N (a) = 1$ if $\gcd(a, N) = 1.$ Prove that ...
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1answer
19 views

Algebraic Number Theory - Rings of Integers

Let $K$ be a number field. Then the ring of integers of $K$ is defined as $K \bigcap \mathbb{B} = \mathcal{O}_{K}$. An integral basis of $K$ is defined as a set of elements $b_{1}, b_{2},\dots, b_{n}$ ...
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1answer
28 views

Why is the separable closure of a field in it's completion not complete.

I am currently reading "Algebraic Number Theory" by Neukirch and I am a bit confused by what is here. It is on page 143, it states this: $(K,v)$ is a nonarchimedian valued field and ...
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1answer
45 views

Finite field extension over the rationals need only one generator?

Some book stated (without proof) that in every finite dimensional field over the rationals of dimension $n$, there is an element of degree $n$ (i.e. any field $Q[\alpha_1, ..., \alpha_n]$ is of the ...
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40 views

Finite group of units

I want to show that the group of units of a number field $K$ is finite $\iff K= \mathbb{Q}$ or $K$ is an imaginary quadratic field. I know that the units of a number field are precisely the integral ...
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43 views

In the general number field sieve, do we need to know whether powers of elements in the algebraic factor base divide an element $a+b\theta$?

I'm reading this paper trying to implement the number field sieve. http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.219.2389 Let $\theta$ be the root of some monic ...
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3answers
75 views

Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
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2answers
53 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
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2answers
76 views

quadratic Gauss sum over a power of 2

Is there a general formula for the generalized quadratic Gauss sum defined by $$ G(a,b,c)=\frac{1}{c}\sum_{n=0}^{c-1}e\left(\frac{an^2+bn}{c}\right) $$ where $e(x)=\exp(2\pi ix)$ and $c$ is a power of ...
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1answer
53 views

The root of a monic polynomial with algebraic coefficients.

Let α be a complex number that satisfies α3 + βα2 + γα + δ = 0 β, γ, and δ satisfy cubics with rational coefficients. For example, β satisfies β3 + aβ2 + bβ + c = 0. However, it is not stated that ...
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1answer
21 views

A little doubt about norm of ideals in $O_K$

If $I$ is an integral ideal of $O_K$ for $K=\mathbb{Q}(\theta)$, and $a \in O_K$ then can I say, $N(I)|N(\langle a \rangle) \implies I \ | \ a$
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60 views

Norm and Trace of an element is an integer, then element is an integral?

Let $L/K$ be a finite field extension, and let $\{b_1,b_2,...,b_d\}$ be a basis for $L/K$ My notes define $O_k:=\mathbb{B}\cap K$, where $\mathbb{B}:=\{\alpha$ is algebraic|min poly of $\alpha$ ...
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1answer
46 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
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1answer
46 views

How can I see that $4$ is not a quartic residue?

How can I see that $4$ is no quartic residue, i.e. there is no $t$ such that $t^4 \equiv 4 \mod p$ when $p\equiv 5 \mod 8$?
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1answer
64 views

which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...
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1answer
117 views

Dedekind rings which are UFDs but not PIDs?

I just have a really quick question of an example that I was trying to come up with. Are there any number rings which are UFDs but not PIDs?
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3answers
75 views

Prove some number is algebraic over a field

How do you prove (without calculating the minimum polynomial) that $\sqrt{3}$ + $\sqrt[]{5}$ is algebraic over $\mathbb{Q}$. Also prove that $\left(\mathbb{Q}(\sqrt{3} + \sqrt[]{5}\right):\mathbb{Q}) ...
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1answer
62 views

Quadratic residue modulo $p$ iff quadratic residue module $p^k$

Let $p$ be an odd prime, $a\in \mathbb{Z}$ with $(a,p)=1$. I am trying to show that if $a$ is a square modulo $p$ then it is a square modulo $p^k$. I managed to prove this using an exponential ...
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1answer
66 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 ...
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2answers
42 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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2answers
38 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 ...
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1answer
36 views

Relating Galois groups related via completions

Let $K$ be a number field, and let $K_\mathfrak p$ denote the completion of $K$ at the prime $\mathfrak p$ of $K$. I'm wondering what can be said (if anything useful) that relates the Galois groups ...
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1answer
159 views

Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
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2answers
152 views

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic.

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic. Note: $\alpha \in \mathbb{R}$ I know that if $\alpha$ is algebraic, then there exists some $f(x) \in ...
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1answer
51 views

Questions about the square root of $a$ in $\mathbb{F}_p$.

How to prove that there is a square root of $3$ in $\mathbb{F}_p$ if and only if $p \equiv 1 $ or $11 \pmod {12}$? Thank you very much.
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1answer
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Question about the cyclotomic $\mathbb Z_p$-extension

Let $K$ be a number field and $K_{\infty}/K$ the cyclotomic $\mathbb Z_p$-extension of $K.$ My question is : How to prove that for any prime $\ell$ of $\mathbb Q$ distinct to $p$ does not decompose ...
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1answer
60 views

Ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$

Let $p,q$ be distinct odd primes, and $\omega_p,\omega_q$ primitive $p$-th, $q$-th roots of unity. What is the ring of integers of $\mathbb{Q}(\omega_p,\omega_q)$? The numbers in the ring ...
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1answer
92 views

Interlude on Traces (and another interlude on how bad of a writer Frohlich is)

I'm trying to read through Frohlich's section of Algebraic Number Theory, but this guy really goes out of his way to make sure you don't understand anything. Frohlich is probably the guy Serre is ...
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2answers
172 views

What's a Dual Basis?

If $L/K$ is a finite extension of fields with $v_1, ... , v_n$ a basis, then we have an isomorphism of $K$-modules $L/K \rightarrow Hom_K(L,K)$, where a basis for $Hom_K(L,K)$ is the "dual basis" of ...
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66 views

How to show that either $a^2+5b^2=p$ or $c^2+5d^2=2p$ has integer solutions for all prime $p$ with $(\frac{-5}{p})=1$

How to show that either $a^2+5b^2=p$ or $c^2+5d^2=2p$ has integer solutions for all prime $p$ with $(\frac{-5}{p})=1$ would the fact that $\mathbb{Z}[\sqrt{-5}]$=$\mathcal{O}\cap ...
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1answer
66 views

How to show that every prime ideal 0f R contains a unique prime of $\mathbb{Z}$

Let $K$ be a number field with the ring of integers of $R$. How to show that every prime ideal of $R$ contains a unique prime of $\mathbb{Z}$. I have no idea, could you please help.
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1answer
337 views

Norm of the product of two regular ideals of an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank ...
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2answers
128 views

irreducible polynomial in $Z_p [x]$

Let $p$ be a prime integer. prove that the following are equivalent (a) $p$ is a prime in $Z[i]$ (b) $x^2 +1$ is irreducible in $Z_p [x]$ What I know is that if p is a prime in $Z[i]$, $p$ is ...
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1answer
73 views

work out the value of a - b from the identity $ax+18=2(x-b)$

How do I solve the following question? You are given the algebraic identity: $ax+18=2(x-b)$ Work out the values of $a-b$
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1answer
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Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
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1answer
46 views

Proving that a ring has only finitely many prime ideals

Let $D$ be a Dedeking domain, $\mathfrak{i}$ a nonzero ideal of $D$ and let $B=D/\mathfrak{i}$ be the quotient ring. Then $B$ is a noetherian ring, and every prime ideal of $B$ is maximal. I have ...
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1answer
208 views

Cyclotomic euclidean number fields

I´m here because I want to repeat my question about the norm-euclidean algorithm in a particluar cyclotomic integer ring. Let $L=\mathbb{Q}(\zeta_{32})$ and $A=\mathbb{Z}[\zeta_{32}]$. On the page ...
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1answer
208 views

Norm of an ideal

Would someone attempt to give me a simple explanation of how to compute the norm of an ideal. I like the definition |$O/a$| but find it difficult to apply. Perhaps the "determinant version" lends ...
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1answer
37 views

The order of $P(K)/P^+(K)$ in a quadratic number field

Let $K/\mathbb{Q}$ be a quadratic extension. Let $P(K)$ be the group of principle fractional ideals of $\mathcal{O}_K$. Let $P^+(K)$ be the subgroup of principle fractional ideals with generator with ...