Questions related to the algebraic structure of algebraic integers

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2
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0answers
174 views

The number of genera of binary quadratic forms of given discriminant

Is the following proposition true? If yes, how do we prove it? Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Let $\Phi_1,\dots,\Phi_{\mu}$ be ...
3
votes
0answers
237 views

The narrow class group of an order of a quadratic field and the genera of binary quadratic forms

Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$. Let $F = ax^2 + bxy + cy^2 \in \mathfrak{F}$. Let $\alpha = \left( \begin{array}{ccc} p & q \\ r & s \end{array} ...
8
votes
2answers
581 views

Bijection between an ideal class group and a set of classes of binary quadratic forms.

Let $F = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. If $D$ is not a square integer and gcd($a, b, c) = 1$, we say $ax^2 + bxy + ...
0
votes
1answer
376 views

Relations between fractional ideals of an order of a quadratic number field and binary quadratic forms

Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$. Let $F = ax^2 + bxy + cy^2 \in \mathfrak{F}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. It is easy to see that $D ...
6
votes
1answer
472 views

A binary quadratic form and an ideal of an order of a quadratic number field

Let $F = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. If $D$ is not a square integer and gcd($a, b, c) = 1$, we say $ax^2 + bxy + ...
1
vote
1answer
349 views

Discriminant of a binary quadratic form and an order of a quadratic number field

Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D = b^2 - 4ac$ be its discriminant. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Conversely ...
1
vote
1answer
110 views

Extension of rings of integers always locally free

In his answer to this question, Andrea claims that if $A \subset B$ is an extension of rings of integers of number fields, $B$ is locally free over $A$. How can one prove this? Furthermore, I am ...
4
votes
1answer
242 views

A prime in a separable extension splits completly iff it does so in the galois closure.

I just did the following exercise out of Neukirch's Algebraic Number Theory: "A prime ideal p of K is totally split in the separable extension L|K iff it is totally split in the Galois closure N|K of ...
1
vote
0answers
423 views

Transforming root-equations into polynomials

Let's define special polynomials as polynomials in $\mathbb{Q}[X]$, where we allow to make roots, too. Examples: $\sqrt{X^4+1}$, $\sqrt[3]{X}+\sqrt{X+1}$, $\sqrt{X+\sqrt{X+1}}$ How can I transform a ...
3
votes
1answer
108 views

If $S$ is integral over integrally closed $R$, can $\mathfrak{a}S$ be principal without $\mathfrak{a}\triangleleft R$ being principal?

Let $R\subset S$ be integral domains, with $R$ integrally closed in its field of fractions, and $S$ integral over $R$. Suppose that the fraction field of $S$ is a finite Galois extension of the ...
8
votes
2answers
422 views

Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
10
votes
1answer
238 views

Connection between ramification in number fields and Clifford theory

Consider algebraic number fields $\mathbb{Q} \subseteq K \subseteq L$ with rings of integers $\mathbb{Z}\subseteq \mathcal{O}_K \subseteq \mathcal{O}_L$. If $0 \neq \mathfrak{p} \trianglelefteq ...
2
votes
1answer
147 views

The different of a Galois extension of an algebraic number field

Since the ramification indices of a prime are the same in a Galois extension, the following proposition is likely to be true. If it is true, how do we prove it? Proposition Let $K$ be an algebraic ...
4
votes
2answers
275 views

The primes $p$ of the form $p = -(4a^3 + 27b^2)$

The current question is motivated by this question. It is known that the number of imaginary quadratic fields of class number 3 is finite. Assuming the answer to this question is affirmative, I came ...
8
votes
1answer
443 views

The ring of integers of a number field is finitely generated.

For a number field $K$, we define the ring of integers of $K$ to be $$\mathcal{O}_K:=\{x\in K\big|\ (\exists f\in\mathbb{Z}[X])(f\ \text{ is monic and } f(x)=0)\}.$$ Is there any easy way to see from ...
4
votes
1answer
493 views

Hilbert class field of a quadratic field whose class number is 3

Is the following proposition true? If yes, how would you prove this? Proposition Let $f(X) = X^3 + aX + b$ be an irreducible polynomial in $\mathbb{Z}[X]$. Let $d = -(4a^3 + 27b^2)$ be the ...
20
votes
2answers
1k views

Unramification of a prime ideal in an order of a finite Galois extension of an algebraic number field

Is the following proposition true? If yes, how would you prove this? Proposition Let $K$ be an algebraic number field. Let $L/K$ be a finite Galois extension. Let $A$ and $B$ be the rings of ...
6
votes
2answers
321 views

Hilbert class field of $\mathbb{Q}(\sqrt{65})$

Let $K = \mathbb{Q}(\sqrt{65})$. Let $L = \mathbb{Q}(\sqrt{5}, \sqrt{13})$. Is $L$ the Hilbert class field of $K$? If yes, how would you prove this?
1
vote
2answers
307 views

Discrete valuation ring associated with a prime ideal of a Dedekind domain

Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$. Let $P$ be a non-zero prime ideal of $A$. Let $v_P$ be the valuation of $K$ with respect to $P$. Then the localization $A_P$ of ...
6
votes
1answer
95 views

Matrix groups generated by translation and inversion in the unit sphere

Let $\alpha$ be algebraic over $\mathbb{Q}$, and consider the subgroup $G$ of $\mathrm{SL}_2(\mathbb{C})$ generated by inversion in the unit sphere and translation by $\alpha$. That is, consider ...
6
votes
1answer
182 views

Two corollaries in Lang's Algebraic Number Theory.

I'm having difficulty understanding the relationship between two corollaries in Lang's Algebraic Number Theory, on page 16 for those with the book. They can also be found in his Algebra. The first ...
8
votes
2answers
920 views

How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
5
votes
1answer
195 views

Pell type equation: $x^2-py^2=a$

Let $p=4k+1$ be a prime number such that $p=a^2+b^2$, where $a$ is an odd integer.Prove that the equation $$x^2-py^2=a$$ has at least a solution in $\mathbb{Z}$.
2
votes
1answer
128 views

Can we descend field extensions of prime degree of number fields to number fields of the same degree

Let $K$ be a number field and let $p$ be a prime number. Let $L$ be a degree $p$ field extension of $K$. Does there exist a degree $p$ field extension $M$ of $\mathbf{Q}$ such that ...
18
votes
0answers
422 views

Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's ...
13
votes
5answers
704 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
2
votes
1answer
99 views

Technical question on integral ring extensions

Let $A$ be an integral domain, integrally closed in its field of quotients $K$ and let $L$ be a finite Galois extension of $K$ with group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $p$ be ...
5
votes
1answer
138 views

Quadratic field such that a certain finite set of primes split

Given a finite set $S$ of primes, is it possible to find an imaginary quadratic field $K$ such that all primes in $S$ are split completely in $K$?
2
votes
2answers
232 views

Rational points on singular curves and their normalization

Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable. Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?
3
votes
3answers
2k views

Pre-requisites needed for algebraic number theory

I acknowledge my limited knowledge of abstract algebra(My background comprising groups and subgroups from Herstein's Topics in Algebra is hardly worth mentioning) .And yet, I confess I really like ...
3
votes
1answer
239 views

On the class number of a cyclotomic number field of an odd prime order

Is the following proposition true? If yes, how would you prove this? Proposition Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = ...
3
votes
3answers
128 views

Proving $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$, given that $\theta^3 + 11\theta - 4 = 0$

As the title says, given that $\theta^3 + 11\theta - 4 = 0$, I'm trying to prove that $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$. I know that $x^3 + 11x -4$ ...
2
votes
1answer
272 views

A certain problem concerning a Hilbert class field

Is the following proposition true? If yes, how would you prove this? Proposition Let $k$ be an algebraic number field. Let $K$ be a finite abelian extension of $k$. Suppose every principal prime ...
-1
votes
2answers
331 views

On a certain criterion for unramification of an abelian extension of an algebraic number field

Probably the following proposition can be proved using class field theory. But I don't know how. My question: Is the following proposition true? If yes, how would you prove this? Proposition Let $K$ ...
-1
votes
1answer
311 views

Complete splitting of a prime ideal in a certain abelian extension of an algebraic number field

Is the following proposition true? If yes, how would you prove this? Proposition Let $k$ be an algebraic number field. Let $K$ be a finite abelian extension of $k$. Suppose every principal prime ...
0
votes
1answer
683 views

Ramified primes in a cyclotomic number field of a prime power order

Let $l$ be a prime number(even or odd), $n \geq 1$ an interger. Let $\zeta$ be a primitive $l^n$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Is the following proposition true? If ...
4
votes
3answers
965 views

The group of roots of unity in an algebraic number field

Is the following proposition true? If yes, how would you prove this? Proposition Let $K$ be an algebraic number field. The group of roots of unity in $K$ is finite. In other words, the torsion ...
5
votes
6answers
344 views

How to prove that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer

As the title says, I'm trying to show that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer. I suppose there's probably some heavy duty classification theorems that give one line ...
2
votes
1answer
356 views

How to compute efficiently the norm of a cyclotomic integer

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\alpha \in A$. How ...
5
votes
2answers
524 views

Question about $p$-adic numbers and $p$-adic integers

I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks. Let $p$ be a prime. Then we define the ring of $p$-adic integers to ...
-5
votes
1answer
295 views

Real units of the cyclotomic number fields of an odd prime order

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\epsilon$ be a unit ...
-2
votes
1answer
1k views

The group of roots of unity in the cyclotomic number field of an odd prime order

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Is it true that any root of unity in $K$ is of the form $\pm\zeta^k$ where ...
0
votes
1answer
120 views

On a certain property of units in the cyclotomic number field of an odd prime order

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $\epsilon$ be a unit ...
1
vote
2answers
241 views

Is there a purely imaginary unit in the cyclotomic number field of an odd prime degree?

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. My question: Is there a ...
1
vote
1answer
76 views

On a certain property of the different of an extension of an algebraic number field of a prime relative degree

Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $L$ be an extension of $K$ of a prime degree $p$. Let $B$ be the ring of integers in $L$. Let $\mathfrak{D}_{L/K}$ be ...
1
vote
1answer
106 views

The different of a cyclic extension of an algebraic number field of a prime degree

Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $L$ be a cyclic extension of $K$ of degree $l$, where $l$ is a prime number. Let $B$ be the ring of integers in $L$. ...
1
vote
2answers
68 views

Selfconjugate prime ideal of a cyclic extension of an algebraic number field of prime degree.

Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $L$ be a cyclic extension of $K$ of degree $l$, where $l$ is a prime number. Let $B$ be the ring of integers in $L$. ...
2
votes
1answer
742 views

Maximal real subfield of $\mathbb{Q}(\zeta )$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. My question: Is the following proposition true? If yes, how would you ...
4
votes
0answers
238 views

Is an algebraic integer all of whose conjugates have absolute value 1 a root of unity? [duplicate]

Possible Duplicate: Are all algebraic integers with absolute value 1 roots of unity? Let $\alpha$ be an algebraic integer. Suppose that all the roots of its minimal polynomial have absolute ...
1
vote
1answer
134 views

Special units of the cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $k$ be a rational ...