Questions related to the algebraic structure of algebraic integers

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Is the following a number field?

Is the field obtained by adjoining all the cube roots of $-3$ to $\mathbb Q$ a number field ? The cube roots of $-3$ are: $-\sqrt[3]{3},\sqrt[3]{3}e^{\frac{i\pi}{3}}, ...
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Prime ideals lying above in $\mathbb{Q}(\sqrt{-5})$

I'm really struggling to understand the concept of prime ideals lying above and below a given prime ideal. For example taking the extension $\mathbb{Q}(\sqrt{-5})\big/\mathbb{Q}$, how do we know $(2, ...
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1answer
26 views

Number rings as free module over base ring

Let $K \subset L$ be number fields and $\mathcal{O_K}, \mathcal{O_L}$ the corresponding rings of algebraic integers. Further let dimension$(L/K)$ = n as a vector space. If $K$ is a PID, then ...
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Determine the splitting field of $x^n - 1$ over $\mathbb{Q}_p$ [closed]

Given a prime number $p$, how can one determine the splitting field of $x^n - 1$ over $\mathbb{Q}_p$ the p-adic number field? The case for $\mathbb{Q}$ is well known, so I am thinking that the ...
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3answers
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Find all Gaussian integers $α, β, γ$ such that $αβγ = α + β + γ = 1$

I tried to solve for this by assuming $α=a+bi$, $β=c+di$, and $γ=e+fi$, and explicitly solving this by equal $a+c+e=1$, $b+d+f=0$, and similarly for $αβγ=1$. Is there any other easier approach for ...
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Stuggling to understand ideal powers

In my current algebraic number theory course we have defined the multiplication of 2 ideals as the smallest ideal containing all products of elements of both, [i.e: let I and J be ideals of a ring ...
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14 views

The “good” singularities of a local model?

In the theory of Shimura Varieties you want to construct a model over the ring of integers of the reflex field of the Shimura variety. You want it to be flat and have "good" singularities. This ...
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1answer
25 views

Algebraic element proof

Definition-Lemma: Let $F$ be a subfield of a field $L$. An element $a\in L$ is called algebraic over $F$ if one of the following equivalent conditions hold: $f(a)=0$, for an non-zero polynomial ...
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1answer
53 views

Roots of unity of an odd degree number field

I want to show that a number field of odd degree contains only $2$ roots of unity. The only information I really have regarding this that I think is relevant is that the group of units ...
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1answer
30 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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1answer
54 views

Small integral representation as $x^2-2y^2$ in Pell's equation

Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. ...
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1answer
47 views

Checking a fundamental unit of a real quadratic field

I just want to check whether I have got the fundamental unit of a certain real quadratic field, but I can't find how. For instance, if I am working in $\mathbb{Q}(\sqrt{2})$ then ...
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3answers
55 views

Condition on ideal of ring of integers being prime

Let $K=\mathbb{Q}(\sqrt{d})$ for $d$ square free integer and let $p$ be a rational prime such that $p$ does not divide $2d$. Prove that $p\mathcal{O}_K$ is a prime ideal $\iff x^2\equiv_p d$ has no ...
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0answers
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Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring ...
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1answer
43 views

Relationship between class number and Legendre symbol

Suppose we have a prime $p\equiv 3\mod 4$ and $p>3$ with the property that for all primes $q<p/4$, we have that $\left(\frac{q}{p}\right)=-1$. I believe that in this case it is true that the ...
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2answers
55 views

Quadratic Integers in $\mathbb Q[\sqrt{-5}]$

Can someone tell me if $\frac{3}{5}$, $2+3\sqrt{-5}$, $\frac{3+8\sqrt{-5}}{2}$, $\frac{3+8\sqrt{-5}}{5}$, $i\sqrt{-5}$ are all quadratic integers in $\mathbb Q[\sqrt{-5}]$. And if so why are they in ...
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0answers
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Absolute values induced by embeddings (after Lang's “Algebraic number theory”)

I am now reading Lang's "Algebraic number theory" (http://gen.lib.rus.ec/book/index.php?md5=9D32C9B248831979ADE79FACDC40129B) and I am having problems with understanding the statement of the theorem 2 ...
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1answer
91 views

Artin conductor of a character and factorisation through $(\mathbb{Z} / N\mathbb{Z})^{*}$

This is from Serre's paper on modular representations of degree $2$ of $Gal(\bar{\mathbb{Q}} : \mathbb{Q} ) $. We consider a representation $\rho : Gal(\bar{\mathbb{Q}}:\mathbb{Q}) \rightarrow ...
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1answer
30 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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25 views

Lucas's Cyclotomic Formula

There are 2 well-known formulas involving Cyclotomic polynomials, which can be described roughly as writing $\phi_n$ as norms of elements in some quadratic extension. They appear in wikipedia and ...
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1answer
29 views

Splitting a Set of Consecutive Cubes into Equal Subsets

Is it possible to split the set of 14 consecutive cubes $1^3,2^3,\ldots,14^3$ into two subsets of equal sums? There has to be a more efficient approach than brute force, right? Because with brute ...
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2answers
33 views

Factorization of polynomial in a complete field

Let $k$ be a finite extension of $\mathbb{Q}$ and $|\cdot|$ an absolute value on it (either Archimedean or not). Let $L$ be the completion of $k$ with respect to this value, and take any irreducible ...
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Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid?

$$5 = (-1) \left( \frac{3 - \sqrt{29}}{2} \right) \left( \frac{3 + \sqrt{29}}{2} \right)$$ or $$5 = \left( \frac{7 - \sqrt{29}}{2} \right) \left( \frac{7 + \sqrt{29}}{2} \right)?$$ ...
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1answer
41 views

Isomorphism of quotient rings

In a course on algebraic number theory, the lecturer says $$\mathcal{O}_K\cong \mathbb Z\left[\frac{1+\sqrt d}{2}\right] \cong\frac{\mathbb Z[x]}{\left( x^2-x-\frac{d-1}{4} \right)}.$$ This ...
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2answers
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For which values of $d<0$ , is the subring of quadratic integers of $\mathbb Q[\sqrt{d}]$ is a PID?

The "integers" of quadratic field $\mathbb Q[\sqrt{d}]$ , for a squarefree integer $d$ , forms an integral domain . I know that for $d<0$ , the quadratic integers of the quadratic number fields ...
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1answer
21 views

Finding an integral basis for a lattice defined in terms of an equation modulo p

Let p be a prime number, u relatively prime to p, and $\Lambda := \lbrace (a, b) \in \mathbb{Z}^2 : b \equiv au$ (mod p)$\rbrace$. How then can I find an integral basis $v_1, v_2$ for $\Lambda$? I ...
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1answer
38 views

If $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ?

If $a,b \in \mathbb Z$ are such that g.c.d.$(a,b)=1$ and if $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ? I cannot use ...
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Prime ideals contained in the union of almost all prime ideals

I am reading the proof of the long exact sequence involving $S$-class groups and $S$-units in Neukirch Algebraic Number Theory, Chapter I, Prop. 11.6, which states the following canonical sequence is ...
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1answer
29 views

Liouville's and Roth's theorems for complex algebraic numbers

Liouville's theorem says that If $\alpha$ is an irrational number which is the root of a polynomial $p$ of degree $d > 0$ with integer coefficients, then there exists a real number $C > 0$ ...
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Field extension of K with unique factorization?

When solving Diophantine equations, often I pass to a number field $K$ and hope that the algebraic integers $O_K$ have unique factorization. Suppose that $O_K$ is not a UFD. Is it possible that there ...
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1answer
39 views

Prime ideal in the ring of integers of the number field $\mathbb{Q}(x)$ with $x^{3}=2$

In an exercise of the book Algebraic Theory of Numbers by Samuel, one must show that--in the integer ring $\mathcal{O_k}$ of the number field extension $\mathbb{Q}(x)$ where $x^{3}=2$--the ideal ...
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2answers
56 views

Sums of Consecutive Cubes (Trouble Interpreting Question)

Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums. Any suggestions on what techniques should be used to ...
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3answers
69 views

Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$

I am working with the ideals $\mathfrak{p}=\left<2,1+\sqrt{-5}\right>, \mathfrak{q}=\left<3,1+\sqrt{-5}\right>, \mathfrak{t}=\left<3,1-\sqrt{-5}\right>$ and I am trying to prove that ...
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2answers
51 views

Ring homomorphism takes discriminant to discriminant

Let $R[x] \xrightarrow{\sigma} S[x]$ be a ring homomorphism where $R,S$ are integral domains of characteristic $0$. Is it true that for any monic polynomial $f(x) \in ...
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1answer
26 views

If the limit of a sequence of algebraic integers is algebraic, does it need to be an algebraic integer?

Consider a sequence $\{\alpha_n\}$ of algebraic integers and let $\alpha = \lim_{n \to \infty} \alpha_n$, where the limit is taken with respect to the usual absolute value in $\mathbb{C}$, and suppose ...
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1answer
26 views

Finding powers of prime ideals from its generators and understanding generator notation

I am trying to understand ideal notation with pointed brackets and how to use it. For instance, if I had an ideal $\mathfrak{a}=\left<2,1+\sqrt{-5}\right>$, where $2$ and $1+\sqrt{-5}$ are its ...
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1answer
28 views

Norm of an ideal is finite

I want to show that the norm $N_{K/\mathbb Q}(\mathfrak{a})$ of $\mathfrak{a}$ a nonzero integral ideal of a number field $K$ is finite, and so $N_{K/\mathbb Q}(\mathfrak{ab})=N_{K/\mathbb ...
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1answer
28 views

Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum ...
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3answers
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Determinant of a Vandermonde matrix of roots of monic polynomial with integer coefficients

Let $p(x)=\sum_{i=1}^n a_ix^i$ with $a_i$ an integer for all $i$ and $a_n=1$ such that $p(x)$ has only real roots, and let $\lambda_1,\ldots,\lambda_n$ be the $n$ roots of this polynomial. Then the ...
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Elementary solution to the Mordell equation $y^2=x^3+9$?

I've recently been wondering how to solve the equation of mordell for k=9, namely: (y^2=x^3+9). It reduced to solving the Thue equation (|a^2-2b^3|=3).Interestingly, the equation has several ...
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Ramanujan conjecture and Langlands program

In the article http://www.thehindu.com/sci-tech/science/the-legacy-of-srinivasa-ramanujan/article2746988.ece, it was mentioned that "This conjecture, later called Ramanujan's conjecture, came to ...
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5answers
150 views

Why is quadratic integer ring defined in that way?

Quadratic integer ring $\mathcal{O}$ is defined by \begin{equation} \mathcal{O}=\begin{cases} \mathbb{Z}[\sqrt{D}] & \text{if}\ D=2,3\ \pmod 4\\ ...
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1answer
52 views

Number of finite extensions of $p$-adic number field of given degree $n$

Let $p$ be a prime number, $\mathbb{Q}_p$ the $p$-adic number field. We fix an algebraic closure $\Omega$ of $\mathbb{Q}_p$. Any algebraic extension of $\mathbb{Q}_p$ is assumed to be a subfield of ...
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1answer
52 views

Motivation for the definition of the Artin Conductor of a representation

I'm trying to figure out what the Artin Conductor of a representation is using Chapter IV.$2\text-3$ of Serre's Local Fields, and I'm struggling to understand the motivation behind its definition. ...
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35 views

proving Fermat's theorem on $p = x^2 + 3y^2$

Here is a modern proof from the notes primes presented by quadratic forms. We are interested in $p = x^2 + 3y^2$ so we would like to have something like: $$ p = (x + y\sqrt{-3})(x - y\sqrt{-3}) = ...
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1answer
39 views

How many positive solutions are there (Positive 3 tuples)?

I want to find how many positive solutions for the Diophantine equation $4x + 2y + 5z = 100$ I found a particular solution $(x,y,z) = (50,-50,0)$ then I found a general solution (basis) $s(-2,-1,2) + ...
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1answer
9 views

prime ideal of integral closure on the decomposition iff lies above the prime ideal of the ring

I'm having troubles proving the following proposition. In every reference I read, they mark this proposition as "clear" or "trivial", but I am unable to prove it. Some help? Let $A$ be a Dedekind ...
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12 views

Number of isotropic vectors of a hermitian form

Good evening, I have a question about isotropic vectors in hermitian spaces and I hope someone can help me out. Let K be a local non-dyadic field and $\pi$ a prime element (so 2 is a unit). Let $h$ ...
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1answer
19 views

Properties of the exponent function attached to a nonzero prime ideal in a Dedekind domain

I want to prove properties of $v_\mathfrak{p}$, which I have been told is: "the exponent function attached to a nonzero prime ideal $\mathfrak{p}$ that maps a given nonzero fractional ideal to the ...