Questions related to the algebraic structure of algebraic integers

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3
votes
1answer
129 views

Ring of integers of $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$

Let $p,q$ be distinct prime numbers $\ge 5$ such that $pq^2 \not\equiv 1\mod9$. Let $K=\Bbb Q[u]$ where $u=\sqrt[3]{p^2q}$, and $A$ be the ring of integers of $K$. I have shown that $u,v=pqu^{-1}\in ...
4
votes
1answer
58 views

Gauss sums ray class group

In Neukirch's book, page 503, in remark 1 he says that He gives a reference for that. I'm not able to get it. Could someone at least draw the shape of the indicated Gauss sum, beause I have ...
2
votes
1answer
631 views

How do we find prime ideals of a ring of integers of a number fileld?

For example for $F=Q(\sqrt{-5})$. the ring of integers of $F =Z[\sqrt{-5}]$.(since $-5\equiv3 \pmod 4$) but how can we determine prime ideals of this? and another problem is the corresponding ...
2
votes
0answers
169 views

Norm of a $\mathbb{Z}/n\mathbb{Z}$-algebra is surjective

Let $n$ be positive integer and $R = \mathbb{Z}[\alpha]$ be the ring of integer of a quadratic number field where $\alpha$ is the root of the quadratic polynomial $X^2 -uX + v \in \mathbb{Z}[X]$, such ...
8
votes
1answer
382 views

Principal ideal domain not euclidean

Can anyone give an example of a principal ideal domain that is not Euclidean and is not isomorphic to $\mathbb{Z}[\frac{1+\sqrt{-a}}{2}]$, $a = 19,43,67,163$? I believe it is conjectured that no ...
1
vote
1answer
55 views

definition of discriminant and traces of number field.

Let $K=\Bbb Q [x]$ be a number field, $A$ be the ring of integers of $K$. Let $(x_1,\cdots,x_n)\in A^n$. In usual, what does it mean $D(x_1,\cdots,x_n)$? Either $\det(Tr_{\Bbb K/ \Bbb Q} (x_ix_j))$ or ...
5
votes
1answer
281 views

Reduction modulo $p$ in number fields

For every prime number $p$, there exist a map $$f:\mathbb{P}^n(\mathbb{Q})\to\mathbb{P}^n(\mathbb{F}_p)$$ defined by: for $P\in \mathbb{P}^n(\mathbb{Q})$, we can find a unique tuple ...
1
vote
1answer
178 views

Ring of integers of $f(X)=X^3+X^2-2X+8$ is principal

Let $K=\Bbb Q[x]$ be a cubic number field with $x$ be a root of $f(X)=X^3+X^2-2X+8$. I want to show that $A$, the ring of integers of $K$ is principal. What I have shown is that $A=\Bbb Z+\Bbb ...
3
votes
0answers
66 views

Tensor product of fields and ramification theory

In the wikipedia page: http://en.wikipedia.org/wiki/Tensor_product_of_fields They write: "For example if one adjoins √2 to the rational field ℚ to get K, and √3 to get L, it is true that the field M ...
2
votes
1answer
103 views

Image of the idele class group and its subgroup of idelic norm 1

[Sorry if the title isn't specific, it was too long.] My question is: Why does $J_{K}/J_{K}^{1}\cong %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ imply that $J_{K}/K^{\ast }$ and ...
1
vote
3answers
99 views

How to show $\mathbb{Z}[w]/(2,w) \simeq \mathbb{Z}_2$?

Let $\mathbb{Z}[w]=\mathbb{Z}[\frac{1+\sqrt{-15}}{2}]$ be the quadratic integers. I want to show that $\mathbb{Z}[w]/(2,w) \simeq \mathbb{Z}_2$. It seems very clear, but how can I show the ...
1
vote
1answer
106 views

ideal and ideal classes in the ring of integers.

(I'm studying Pierre Samuel's Algebraic theory of numbers.) Let $K$ be a number field of degree $n$, $d$ the absolute discriminant of $K$, and $\mathfrak{a}$ a nonzero integral ideal of $K$. ...
4
votes
1answer
94 views

Units of the quotient of an order

Let $n$ be a positive integer and $R$ be an order in a imaginary quadratic number field such that $disc(R)$ is prime to $n$. Further suppose that for every prime $p$ dividing $n$, $p$ is inert in $R$. ...
13
votes
2answers
1k views

What does the discriminant of an algebraic number field mean intuitively?

If $E/F$ is a finite extension of fields and $\alpha_1,\ldots, \alpha_n$ is a basis of $E/F$, the discriminant of $\{\alpha_1,\ldots, \alpha_n\}$ is $$\det(\operatorname{Tr}_{E/F}(\alpha_i\alpha_j))$$ ...
2
votes
3answers
166 views

Why is $\mathbb{Z}[\alpha ]$ not finitely generated as an $\mathbb{Z}$-module?

Assume that $\alpha \in \mathbb{C}$ is an algebraic number which is not an algebraic integer. My question is why $\mathbb{Z}[\alpha]$ is not finitely generated as an $\mathbb{Z}$-module. Clearly there ...
2
votes
1answer
30 views

How to calculate $\sum_{i= 0}^{k-1}\left \{ \frac{ai+b}{k} \right \}$

How to calculate: $$F=\sum_{i= 0}^{k-1}\left \{ \frac{ai+b}{k} \right \}$$ where $a,b \in \mathbb Z$ and $(a,k)=1$; $\left \{ \frac{ai+b}{k} \right \} =$ fraction part of $\frac{ai+b}k$ Such problem ...
7
votes
3answers
415 views

Tensor product of a number field $K$ and the $p$-adic integers

In a paper by J. Jones and D. Roberts (http://math.la.asu.edu/~jj/localfields/database.pdf) we are introduced to an isomorphism $K \otimes \mathbb{Q}_p \cong \prod\limits_{i=1}^g K_{p,i}$, where $K$ ...
3
votes
0answers
43 views

notation for invariation

Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
1
vote
1answer
137 views

ideals in rings of algebraic integers are finitely generated

I am trying to write about rings of algebraic integers $\mathcal{O}_K$ in a number field $K$ without introducing to much field theory. I want to show that these rings are Dedekind. First of all I want ...
3
votes
1answer
454 views

Frobenius element in cyclotomic extension

Let $K=\mathbb{Q}(\zeta_m)$. Then if $p\nmid m$ is any odd prime, how i can show that Frobenius map is $(p,K/\mathbb{Q})(\zeta_m)=\zeta_m^p$. We know, if $P$ is a prime above $p$ ...
4
votes
1answer
380 views

Integral basis for a number field

I need some help in solving the following problem: Suppose $K$ is a number field and $K=\mathbb{Q}(\theta)$ where $\theta\in\mathfrak{O}_K$, the ring of integers of $K$. Now among the elements in ...
0
votes
2answers
83 views

Prove that $\sqrt{1+\pi^2}, \hspace{2pt}\pi-\sqrt\pi,\hspace{2pt} \pi^2+\pi+\sqrt{1+2\pi}$ are not algebraic

Considering the fact that $\pi$ is not an algebraic number, I need to prove these expressions are not algebraic : $$\sqrt{1+\pi^2},\hspace{10pt} \pi-\sqrt\pi,\hspace{10pt} \pi^2+\pi+\sqrt{1+2\pi}$$ ...
5
votes
3answers
821 views

Find generator of principal ideal

The ideal $(9, 2 + 2\sqrt{10})$ of $\mathbb{Z}[\sqrt{10}]$ is a principal ideal; it is generated by $1+\sqrt{10}$. This is easy enough to check once it's been found, but can anyone tell me some way ...
2
votes
1answer
58 views

Do $R'$ integral over $R$ imply $\operatorname{Frac}(R')$ algebraic over $\operatorname{Frac}(R)$?

This is Theorem 6.6 from Janusz, Algebraic Number Fields, it says: Let $R\subseteq R'$ be Dedekind domains with $R'$ integral over $R$ and $\mathfrak{p}$ a nonzero prime ideal of $R$. Suppose ...
6
votes
1answer
2k views

How to calculate the norm of an ideal?

Would someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly. For example, I know that the norm ...
5
votes
1answer
120 views

Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$

I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
6
votes
1answer
100 views

Extension of valuation to the algebraic extension of a number field.

I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to ...
4
votes
1answer
448 views

Irreducible ideal implies prime ideal in Dedekind Domains?

An ideal is irreducible if it can not be written as the finite intersection of strictly larger ideals. In a Noetherian ring every irreducible ideal is primary, but the converse doesn't hold. I wonder ...
9
votes
2answers
137 views

Showing that a real number is an algebraic integer

For what values of $x,y,z\in\mathbb{Z}$, such that $0\leq x,y,z\leq 2, $ the real number $$\alpha:=\frac{1}{3}\left(x+\sqrt[3]{175} \cdot y+\sqrt[3]{245}\cdot z\right)$$ is an algebraic integer i.e. ...
2
votes
1answer
168 views

A calculation of the norm of an ideal

Let $L$ be a number field of degree $n$ over $\mathbb{Q}$ and $\mathfrak{a}$ a non-zero ideal of the ring of integers $\mathcal{O}_L$. Suppose that $X=\{x_1,...,x_n\}$ is a $\mathbb{Z}$-basis of ...
7
votes
1answer
1k views

Proof of Stickelberger’s Theorem

I am having some trouble in understanding the proof of Stickelberger’s Theorem, $\textbf{Theorem :}$ If $K$ is an algebraic number field then $\Delta_K$, the discriminant of $K$, satisfies ...
4
votes
1answer
197 views

Residue fields of gaussian primes

I just started with algebraic number theory and need some help: In the ring of Gaussian integers a prime $p$ with $p=1\bmod 4$ splits into a product of two irreducible elements $(a+bi)(a-bi)$, with ...
2
votes
2answers
65 views

Prime decomposition in ring extensions

Let $R\subseteq R'$ be Dedekind domains, let $\mathfrak{p}$ be a nonzero prime ideal of $R$. Then $\mathfrak{p}R'$ is an ideal of $R'$ and it has a factorization ...
4
votes
2answers
642 views

Class Group of $\mathbb Q(\sqrt{-35})$

As an exercise I am trying to compute the class group of $\mathbb Q(\sqrt{-35})$. We have $-35\equiv 1$ mod $4$, so the Minkowski bound is $\frac{4}{\pi}\frac12 \sqrt{35}<\frac23\cdot 6=4$. So we ...
1
vote
1answer
108 views

Definition: Gauss Sum - Where is the error?

In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct. My intuition says "there is a mistake somewhere". I tried ...
6
votes
2answers
389 views

Show field of fractions is finite extension of $\mathbb{Q}$

Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
7
votes
1answer
304 views

All number fields with absolute value of discriminant $\le 20$

I need to find all number fields with absolute value of discriminant $\le 20$. Using Minkovsky's theorem I understood that it should be quadric or cubic extension. The case of quadric is very ...
6
votes
4answers
223 views

number of solutions to an equation?

Given $x$ and $y$ are multiples of $2$ satisfying $$x^2 - y^2 = 27234702932$$ Find the number of solutions to $x$ and $y$.
0
votes
0answers
54 views

Help me prove the supremum property.

Let $A$ and $B$ be nonempty sets and $f$ be a function from any nonempty set $S$ to subset of real number. Prove that $$\sup_{x \in A} \{ \min \{ \sup_{y \in B} \{ \min \{ f(y) \} \}, f(x) \} \}= ...
1
vote
1answer
422 views

extended Euclidean (xgcd) in quadratic integer rings

Given a discriminant $D < 0$, I have the quadratic imaginary field $\mathbb{K} := \mathbb{Q}(\sqrt{D})$. And the quadratic integer ring is given by $\mathcal{O} = \mathbb{Z} + \mathbb{Z} \frac{D + ...
2
votes
1answer
51 views

The criteria for two abelian extensions to be embedded

Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help. Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if ...
8
votes
3answers
1k views

Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
3
votes
1answer
55 views

If $L\mid K$ is finite not separable then the pairing $\langle x,y\rangle =T_{L\mid K}(xy)$ is degenerate

Let $L$ over $K$ be a finite not separable extension of fields. I want to prove that $$\langle x,y\rangle:=T_{L\mid K}(xy)$$ is degenerate, i.e. there exists a nonzero $x\in L$ such that $\langle x, ...
4
votes
2answers
520 views

How to prove an algebraic integer.

How can you prove that something is an algebraic integer? I have two examples: 1) $\frac{\sqrt{p} + \sqrt{q}}{2}$ , where $p$ and $q$ are integers congruent to $3 \pmod 4$ 2) $\sqrt{\frac{\sqrt{15} ...
12
votes
1answer
283 views

Introduction to the trace formula for people outside number theory

I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula. I am aware that there are "standard references" on the ...
45
votes
5answers
1k views
10
votes
1answer
284 views

Galois representations and normal bases

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
4
votes
2answers
83 views

Let $p$ be an odd prime. $p$ and $(1-\zeta_p)^{p-1}$ are associates in $\mathbb{Z}[\zeta_p]$.

Let $p$ be an odd prime and $\zeta_p$ a primitive $p$th root of unity, that is a $p$th root of unity other than 1. I need to show that $p$ and $(1-\zeta_p)^{p-1}$ are associates in ...
3
votes
1answer
167 views

A proof in Janusz Algebraic Number Field

I can't understand Lemma 6.2 from the Janusz book Algebraic number fields, pag. 26, that says: Let $A\subset B$ be integral domains with $A$ integrally closed and $B$ integral over $A$. If ...
1
vote
1answer
76 views

fraction field of the integral closure

Let $R$ be a domain, $K$ the field of fractions of $R$ and $L$ a finite field extension of $K$. Denote with $R'$ the integral closure of $R$ in $L$. Is it always true that $L$ is the field of ...