Questions related to the algebraic structure of algebraic integers

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5
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1answer
119 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
96 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
406 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
131 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
156 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
185 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
64 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
578 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
96 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
332 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
295 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
220 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
395 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
50 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
462 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
266 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
3answers
1k views
10
votes
1answer
270 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
80 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
148 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
64 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 ...
11
votes
3answers
576 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
2
votes
0answers
39 views

Additive characters of Quotient ring

Suppose $K$ is finite Galois extension of rationals, and let $\mathcal O$ be the ring of integers of $K$, and finally suppose $\mathfrak m$ is an integral ideal of $\mathcal O$, I want to classify ...
3
votes
1answer
54 views

Express in terms of familiar arithmetic functions

How can I express the sumation $$h_k(n)=\sum_{d|n, k|d}\mu (d)$$ in terms of familiar arithmetic functions, where $k\in \mathbb{N} $ is fixed?
1
vote
0answers
50 views

Is it always possible to find primes $p, q$ such that $\left(\dfrac{p}{q}\right)_n=\left(n,\dfrac{p-1}{f}\right)=1$?

I first provide a background, or the context, where this question arises. Skip it if one wants so. Background: In the book The Genus fields of algebraic number fields by Ishida, one finds the ...
6
votes
2answers
449 views

Showing that $\mathbb Q(\sqrt{17})$ has class number 1

Let $K=\mathbb Q(\sqrt{d})$ with $d=17$. The Minkowski-Bound is $\frac{1}{2}\sqrt{17}<\frac{1}{2}\frac{9}{2}=2.25<3$. The ideal $(2)$ splits, since $d\equiv 1$ mod $8$. So we get ...
7
votes
2answers
257 views

Solve: $x^2-py^2=q$

Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers. It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol. ...
5
votes
2answers
241 views

Are algebraic numbers analogous to group elements with finite order?

Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory? I thought this is the case since requiring an algebraic number $\alpha$ to be ...
2
votes
1answer
222 views

fractional ideals in the localization of a Dedekind

I'm reading Janusz, Algebraic number fields, 1973, pag.16-17, where defining a fractional ideal of a Dedekind domain $R$. A fractional ideal of $R$ is a non-zero finitely generated $R$-submodule ...
5
votes
2answers
173 views

Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$

I'm studying for my algebra quals this fall and keep encountering problems like the following: List all the ideals of $\mathbb{Z}[x]/(16, x^3)$. or List all the primes of ...
4
votes
2answers
169 views

Endomorphisms of the multiplicative formal group law

Is there a simple description of the ring of endomorphisms $\mathrm{End}(\mathbb{G}_m)$ of the formal group law $$\mathbb{G}_m(X,Y) = X + Y + XY,$$ at least over a ring of characteristic zero? I'm ...
2
votes
3answers
545 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
5
votes
2answers
581 views

Is the inverse of a fractional ideal still fractional?

Let $R$ be a Dedekind domain, $K$ the field of fractions of $R$, $\mathfrak{m}$ be a fractional ideal of $R$, i.e. a non-zero finitely generated $R$-submodule of $K$. We define ...
6
votes
3answers
1k views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
8
votes
3answers
300 views

Solve $x^3+x \equiv 1 \pmod p$

Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions. We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$ We can prove that $(1)$ cannot ...
5
votes
1answer
447 views

How to determine a Hilbert class field?

I tried to solve the exercise VIII.XX in Number Fields by Marcus. It asks to find the Hilbert class field of $Q(\sqrt m)$ for $m=-6,-10,-21,-30$. And the emphasis of this question is on the first two. ...
1
vote
2answers
118 views

Nonreal units in totally imaginary number fields

Suppose we are given a totally imaginary number field $L$ of degree greater $2$, is it possible that all units of $L$ lie in $\mathbb R$? This is true for almost all quadratic imaginary number fields, ...
8
votes
1answer
1k views

Vague definitions of ramified, split and inert in a quadratic field

Our lecturer defined the following: Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then (1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$ ...
2
votes
1answer
85 views

Euclidean algorithm in lattices in $\mathbb{C}$

Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. I want to prove that, for every $\alpha,\beta\in R,\beta\neq 0$, there exist $\gamma,\delta\in R$ such that $\alpha=\gamma\beta+\delta$, with ...
2
votes
1answer
101 views

How to find $\sup(\{|x-y|_p : x,y\in B(0;r)\})$

Just to clarify the notation and the question: Working in p-adic space $\mathbb{Q}_p$, we have the norm $|x|_p=p^{-ord_p(x)}$ and we define the metric over this space as $d(x,y)=|x-y|_p$. We are ...
2
votes
1answer
70 views

$\mathbb{Q}[\sqrt a_1,\ldots,\sqrt a_k]$ vs. $\mathbb{Q}(\sqrt a_1,\ldots,\sqrt a_k)$

Call an algebraic number polyquadratic if it can be expressed as the sum or difference of a finite number of square roots of rational numbers. (This definition follows Conway-Radin-Sadun rather than ...
1
vote
0answers
45 views

How to measure the failure of Hasse norm theorem?

We know that the failure the unique factorisation is measured by the ideal class-group, that of the local-global principle depends upon the Tate-Shafarevich group. Then I thought: what should be ...
4
votes
1answer
127 views

Finiteness of ideal of given norm

I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$. I know there are "standard proofs" (eg How many elements ...
3
votes
2answers
77 views

Is it true that $\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap \mathbb{Q}$?

I know $\mathbb{Z}_{(p)}\subset \mathbb{Z}_{p}\cap \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at prime ideal $(p)$ and $\mathbb{Z}_p$ is the set of p-adic integers. I ...
4
votes
1answer
68 views

On Selmer's curve

I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
9
votes
1answer
456 views

Intuition for Krasner's Lemma

From Milne's Algebraic Number Theory, we have (he assumes that $K$ is complete with respect to a discrete nonarchimedian absolute value, but I don't know where the discrete part is being used) Let ...