Questions related to the algebraic structure of algebraic integers

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1answer
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How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
2
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1answer
19 views

The torsion subgroup of the group of units $R^{\times}$ is always equal to $\{\pm1\}$

In the ring of integers of the number field of degree $3$, the torsion subgroup of the group of units is always equal to $\{\pm1\}$ I found it here (Proposition $5.12$) only that the subgroup of ...
0
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1answer
24 views

Ideals in the ring of gaussian integers of a given norm

What are the ideals in the ring of gaussian integers of a given norm, (say $20$) ? The ring of integers is $\mathbb Z[i]$ and it is a PID, so any ideal must be principal. If the ideal $I$ is ...
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1answer
33 views

Which one is the ring of integers of $K$; $\mathbb Z[\alpha,\frac{\alpha^2}{2}]$ or $\mathbb Z[\alpha,\frac{\alpha+\alpha^2}{2}]$?

If $f:=T^3-T^2+2T+8$ and $\alpha$ is the unique real root of $f$, If $K:=\mathbb Q(\alpha)$, which one of the following rings are the ring of integers of $K$: $$\mathbb ...
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0answers
28 views

The Nagell-Ljunngren Equation

I have been trying to find the papers of Nagell and Ljunngren, which deal with the equation $$\frac{x^n - 1}{x - 1} = y^2$$ and solve it completely. Many papers cite these papers, but I haven't found ...
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0answers
9 views

Another real quadratic integer ring, Euclidean but not norm-Euclidean, with norm function needing only two adjustments?

I have come to know about $\mathcal O_K$ with $K = \mathbb Q(\sqrt{69})$. The norm function needs to be adjusted to absolute value, as is the case with other real rings, but it also needs to be ...
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0answers
18 views

On the existence of a polynomial in several variables with only one zero

Given an ordered field $\mathbb{K}$, then for all $n>1$ there exists $f\in\mathbb{K}[X_1,\ldots,X_n]\ s.t.\ \mathcal{V}^{\mathbb{K}^n}(f):=\{p\in\mathbb{K}^n:f(p)=0\}=\{(0,\ldots,0)\}$ For ...
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0answers
12 views

Show that the positive units of $K$ form a group isomorphic to $\mathbb Z$

Let $K$ be a cubic field such that $r_1=r_2=1$. Suppose $K$ is imbedded in $ \mathbb R$ ($r_1$ and $r_2$ are the usual notations, $r_1$ denotes the number of isomorphisms $\sigma : K \to \mathbb R$ ...
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3answers
29 views

If α is algebraic over K then all the elements of K(α) are algebraic over K [on hold]

Let $L$ : $K$ be a field extension, $\alpha \in L$ algebraic over $K$. Show that every element of $K(\alpha)$ is algebraic over $K$, where $K(\alpha)$ is the smallest subfield that contains $\alpha$ ...
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2answers
44 views

How to find the degree of an extension field?

How to find the degree of an extension field ? Let $f:=T^3-T^2+2T+8\in\mathbb Z[T]$ and $\alpha$ be the real root of $f$. Why is then $\mathbb Q(\alpha)$ is a number field of degree $3$ ? I've ...
1
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1answer
21 views

Norms and traces example

Example: Let $L=\mathbb{Q}(\sqrt{d})$ be a quadratic extention of $F=\mathbb{Q}$ with square-free integer $d$.Then, $g_{a+b\sqrt{d}}(X)=(X-a-b\sqrt{d})(X-a+b\sqrt{d})=X^2 -2aX+(a^2 -db^2),$ so, ...
2
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1answer
18 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
2
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1answer
27 views

Can prime (Ideals) be ramified/split completely in 'their own field'?

Recently I have come across a few sources where the definitions of primes being ramified or splitting completely do not quite adhere to the way I learned them. I completely understand the 'standard' ...
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0answers
75 views
+50

Diophantine equation $x^2 + 6(y+1)^2 = (y+2)^3$

I'm revising for exams and I've got stuck on an algebraic number theory question. The equation I'm trying to solve is $$ x^2 -2 = y^3, $$ and I was told to rewrite it as $$ x^2 + 6(y+1)^2 = (y+2)^3. ...
1
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1answer
28 views

Calculate the ring of integers of quadratic number field $\mathbb{Q}(\sqrt{d})$ [duplicate]

Calculate the ring of integers of quadratic number field $\mathbb{Q}(\sqrt{d})$ Solution: Let $F$ be an algebraic number field. Then an element $b\in F$ is integral iff it monic irreducible ...
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0answers
35 views

If $a, b$ are transcendental then $a+b$ is transcendental or $ab$ is transcendental [duplicate]

I have to prove the following: If $a, b$ are transcendental then $a+b$ is transcendental or $ab$ is transcendental, or both. I don't have any idea on how to solve this. I already proved this: ...
3
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2answers
67 views

Prove: if $a$ and $b$ are algebraic, then $a + b$, $a - b$ and ab are also algebraic

I have to prove the following: If $a, b \in \mathbb{C}$ and are both algebraic over $\mathbb{Z}$, then: $a + b$ is algebraic over $\mathbb{Z}$ $a - b$ is algebraic over $\mathbb{Z}$ $ab$ is ...
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1answer
24 views

On non-constant multiplicative norms on integral domain and when does the absolute value of the norm is unity implies the element is unit?

Consider $\mathbb Z[\sqrt {d}]$, where $d$ is any non - square integer, define $$N(a + \sqrt d b) = a^2 - db^2 = (a + \sqrt d b)(a - \sqrt d b)$$ as $\mathbb Z \subseteq \mathbb Z[\sqrt {d}]$, so from ...
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0answers
28 views

For what pairs of numbers does the norm function fail as a Euclidean function in $\mathbb{Z}[\sqrt{14}]$?

(By "norm" here I mean "absolute value of the norm)" Are their infinitely many such pairs or is it finite in the sense that its just a few primes that cause the problem (like in that other famous ...
2
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0answers
35 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree ...
2
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1answer
19 views

Discriminant of n algebraic numbers equals $0$ iff the algebraic numbers linearly dependent

Let $K \subset L$ be two number fields with $[L:K] = n$. Let $\{\alpha_i:1 \leq i \leq n\} \subset L$. Then $\operatorname{disc}(\alpha_1 \dots \alpha_n) = 0 \iff \alpha_i$ are linearly dependent ...
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2answers
57 views

Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
2
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1answer
23 views

Finite field extensions - $K(\alpha)$

So I am currently studying Algebraic Number theory and a theorem in the Book states the following: Let $L/K$ be a field extension. Then $\alpha \in L$ is algebraic over $K$ if and only if there is ...
1
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1answer
21 views

Is this condition enough for irreducibility?

Suppose that $f$ is a polynomial in $Z[X]$, such that $f = (X-\alpha_1) ... (X-\alpha_n)$ with $n$ distinct complex, irrational roots. Suppose that $Q[\alpha_i]$ = $Q[\alpha_1, ..., \alpha_n]$ for all ...
4
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1answer
30 views

Quadratic number field which is Euclidean but not norm Euclidean

I am looking for an example of a quadratic field $\mathbb Q[\sqrt d]$ , with $d \equiv 2 $ or $3(\mod 4)$ , whose ring of integers is Euclidean but not norm (http://en.wikipedia.org/wiki/Field_norm ) ...
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1answer
32 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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1answer
22 views

Quadratic Number Fields

I am currently studying elementary Algebraic Number Theory and came across the following statement: Any Number Field $K$ such that $[K:\mathbb{Q}] = 2$ is equal to $\mathbb{Q}(\sqrt{d})$ for a unique ...
4
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3answers
66 views

Are all prime ideals in $\mathbb Z[\sqrt{-5}]$ of the form $\langle c, a + b \sqrt{-5} \rangle$?

Where $a, b, c \in \mathbb Z$? I know that if in an UFD, $\langle c, a + b \sqrt{d} \rangle$ would boil down to a principal ideal. But it seems to me that in $\mathbb Z[\sqrt{-5}]$, for any purely ...
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1answer
13 views

Residue class ring of Dedekind domain

Zariski and Samuel Commutative Algebra Ch V para 7 makes the following statement: If $R$ is a Dedekind domain with an ideal $\mathfrak{a}=\prod_i\mathfrak{p}_i^{n(i)}$ factored into prime ideals, ...
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1answer
57 views
+50

Algebraic number theory topics for undergrads

What are some interesting topics or problems in algebraic number theory which could be presented to students in a first undergraduate algebra course (which covers some elementary number theory, ...
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0answers
27 views

The discriminant of (1,x,x^2) in a cubic field.

Let $K$ be a cubic field such that $K=\mathbb Q[x]$ with $x^3=2$. The discriminant of $\{1,x,x^2\}$ is supposed to be $\begin{vmatrix} 3 & 0 & 0 \\ 0 & 0 & 6 \\ 0 & 6 & ...
3
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1answer
36 views

Reference Request for Minkowski (?) Theorem proof.

I'm looking for a statement, I believe to be due to Minkowski, that says something along the lines of: "For an algebraic number field, $K$, $\exists$ only finitely many prime integers, $p\in ...
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2answers
58 views

$i \notin \mathbb{Q}[\sqrt[4]{2}]$ without using topological properties of $\mathbb{R}$

I can think of two related ways to prove that $i \notin K = Q[\sqrt[4]{2}]$: $K$ is a subset of the real numbers and $i$ is not a real number. $K$ is orderable and no ordered field can contain ...
5
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1answer
28 views

Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
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0answers
40 views
+50

What book or website has nice, colorful diagrams illustrating real quadratic integer rings?

I'm sure you all have seen diagrams, colorful or not, illustrating prime numbers in $\mathbb{Z}[i]$ and $\mathbb{Z}[\omega]$, with some of them helpfully pointing out the inert and splitting primes ...
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2answers
37 views

If $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then $R^{\times}=\mathbb Z\big/6\mathbb Z$

How to show that if $K:=\mathbb Q\left(\sqrt{-3}\right)$ and $R$ is the ring of integers of $K$, then the group of units $R^{\times}=\mathbb Z\big/6\mathbb Z$ Now since $-3\equiv1\mod 4$ the ring ...
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3answers
92 views

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ a number field?

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ of $5$-adic numbers a number field, if yes what is the degree ? To be honest I don't understand the question, what does it mean ...
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1answer
62 views

Number of real embeddings $K\to\overline{\mathbb Q}$

How many real embeddings, $K\to\overline{\mathbb Q}$ with $K=\mathbb Q\left(\sqrt{1+\sqrt{2}}\right)$ are there ? We set $f(x)=x^4-2x^2-1$ and if $\alpha=\sqrt{1+\sqrt{2}}$ then $f(\alpha)=0$. ...
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3answers
63 views

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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3answers
60 views

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, ...
1
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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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0answers
32 views

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 ...
5
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3answers
76 views

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 ...
4
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0answers
28 views

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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0answers
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
24 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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0answers
27 views
2
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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 ...
0
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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 ...
2
votes
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. ...