Questions related to the algebraic structure of algebraic integers

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3
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1answer
18 views

Factor into primes in Dedekind domains that are not UFD's?

Does it make sense to factor numbers into prime numbers in Dedekind domains that are not unique factorization domains? I can't really see how it would make sense.
1
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0answers
35 views

A good book to read with Chapter III of Neukirch's “ANT”

The book Algebraic Number Theory from Neukirch is a beautiful book in ANT, but it still have a serious lack in examples and motivation to the concepts. I've already read the first two chapters of the ...
0
votes
0answers
15 views

$k(\mathfrak p)$ basis for $A / pA$

I'm reading this pdf which is showing that a rational prime $p$ ramifies if and only if it divides the discriminant of its number field $K$. I've come across the following line: Let $p \in \mathbb ...
0
votes
1answer
16 views

I need to show that if $K$ is of characteristic $0$,the algebra $A$ has a primitive generator.

Let $K$ be a field and $A$ a reduced K-algebra of finite dimension over $K$. I need to show that if $K$ is of characteristic $0$, $A$ has a primitive generator (i.e. $A=K[x], x \in A$) I've proved ...
2
votes
1answer
25 views

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
votes
1answer
36 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 ...
1
vote
1answer
47 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 ...
2
votes
1answer
42 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 ...
1
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0answers
35 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
14 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
20 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 $\sigma : K \to \mathbb R$ ...
-4
votes
3answers
42 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 $\alpha$ ...
1
vote
2answers
50 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
vote
1answer
22 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
votes
1answer
19 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
votes
1answer
30 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' ...
4
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0answers
87 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
vote
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
69 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 ...
1
vote
1answer
26 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 ...
3
votes
0answers
29 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
votes
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
votes
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 ...
1
vote
2answers
58 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
votes
1answer
24 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
vote
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
votes
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 ) ...
0
votes
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 ...
0
votes
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
votes
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 ...
0
votes
1answer
14 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, ...
3
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1answer
68 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
votes
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 ...
1
vote
2answers
59 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
votes
1answer
30 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 ...
3
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0answers
46 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 ...
2
votes
2answers
38 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 ...
4
votes
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 ...
1
vote
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 $f(\alpha)=0$. ...
1
vote
3answers
64 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}}, ...
3
votes
3answers
61 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
vote
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
votes
3answers
78 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
votes
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 ...
0
votes
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 ...
0
votes
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 ...