Questions related to the algebraic structure of algebraic integers

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7
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closure of units of number fields in the finite idele topology

Let $K$ be a number field. Denote by $\mathcal O _K^\times$ its rings of units and by $\mathcal O _{K,+} ^\times$ its ring of totally positive units. Further let us denote by $\mathbb A ...
6
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2answers
488 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 $$\det(\operatorname{Tr}_{E/F}(\alpha_i\alpha_j))$$ ...
6
votes
2answers
137 views

Usage of algebraic geometry in understanding the total Galois group of the rational

A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". Could anyone shed some light on this remark, or ...
6
votes
3answers
225 views

What's the difference between $\mathbb{Q}[\sqrt{-d}]$ and $\mathbb{Q}(\sqrt{-d})$?

Sorry to ask this, I know it's not really a maths question but a definition question, but Googling didn't help. When asked to show that elements in each are irreducible, is it the same?
6
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2answers
239 views

Galois extension of $\mathbb{Q}_2$ with Galois group $(\mathbb{Z}/2\mathbb{Z})^3$

I am trying to solve this exercise: Prove that $\mathbb{Q}_2$ has a unique Galois extension F with Galois group $(\mathbb{Z}/2\mathbb{Z})^3$. Compute it's ramification groups. Here is what I have ...
6
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3answers
1k views

Show that a specific ideal is not principal

In some cases, it is quite straightforward to prove that a specific ideal cannot be principal. For example, in the ring of integers of $\mathbb{Q}(\sqrt{-5})$, the ideal $(2,1+\sqrt{-5})$ is not ...
6
votes
4answers
326 views

Reference request for Algebraic Number Theory sources for self-study

I would appreciate any suggestions for book or notes on ANT at a level that I would characterize as advanced beginner. I.e., something assuming familiarity with topics in Dummit & Foote, that is a ...
6
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2answers
283 views

Splitting of primes in an $S_3$ extension

Let $\alpha$ be a root of the polynomial $X^3-X-1$. The polynomial has discriminant $-23$ which is not a square, so the splitting field must have Galois group $S_3$. I would like to figure out the ...
6
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3answers
180 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 $K$ ...
6
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4answers
210 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 $y$.
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votes
2answers
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Local-Global Principle and the Cassels statement.

In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that There is not merely a local-global principle for curves of genus-$0$, but ...
6
votes
3answers
355 views

Why is a number field always of the form $\mathbb Q(\alpha)$ for $\alpha$ algebraic?

My definition of a number field is "a finite extension of $\mathbb Q$". I want to prove that if $L$ is a finite field extension of $\mathbb Q$, then $L = \mathbb Q(\alpha)$ for some $\alpha$ algebraic ...
6
votes
2answers
329 views

The Néron-Tate canonical height on elliptic curves

I have been trying to understand the Néron-Tate global canonical height of algebraic points on elliptic curves. Let $K$ be a number field, $E$ an elliptic curve (over $\mathbb{Q}$, say), and $E(K)$ ...
6
votes
3answers
1k views

What are the integers $n$ such that $\mathbb{Z}[\sqrt{n}]$ is integrally closed?

I was recently reading about integral ring extensions. One of the first examples given is that $\mathbb{Z}$ is integrally closed in its quotient field $\mathbb{Q}$. Another is that ...
6
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1answer
270 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 ...
6
votes
2answers
52 views

Is $(x^2 + 1, y^2 + 1)$ a prime ideal in $\mathbb{Q}[x, y]$?

At first I was looking for a ring homomorphism from $\mathbb{Q}[x, y]$ to a domain with $(x^2 + 1, y^2 + 1)$ as it's kernel, but I could not find one. Now I am thinking: maybe $(x + y)(x - y) = x^2 ...
6
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2answers
153 views

Prove $\sum_{i=1}^{n-k} (-1)^{i+1}\; \cdot \; \tfrac{(k-1+i)!}{k! \cdot(-1+i)!} \; \cdot \; \tfrac{n!}{(k+i)! \; \cdot \; (n-k-i)!}=1$

How do I proof this for n>k? I found this Problem in a book at my universities library, but sadly the book doesn't show a solution. I worked on it quite a long time, but now I have to admit that this ...
6
votes
3answers
315 views

Every finite field is a residue field of a number field

I have read that for any finite field $\mathbb{F}_q$ there exists a number field $F$, and some prime ideal $P$ of $\mathcal{O}_F$ such that $$\mathbb{F}_q \cong \mathcal{O}_F / P.$$ The ring of ...
6
votes
2answers
173 views

Solving the equation by going into a non-UFD

To solve $y^2 + 2 = x^3$ you can factor $(y - \sqrt{-2})(y + \sqrt{-2}) = x^3$ and then check that they are relatively prime and by unique factorization both must be cubes then you can solve it. What ...
6
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2answers
78 views

References for elliptic curves over schemes

As in the title, I want some references about theories for elliptic curves over rings(not fields) or over schemes. I heard that behaviours(?) of such elliptic curves are not as simple as elliptic ...
6
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1answer
299 views

Lattice of Gauss and Eisenstein Integers

Z is a 1D lattice Gaussian and Eisenstein integers are 2D lattices But the golden integers (for example) are dense on the real line. Are there rings of integers which have 3D, 4D, ... lattices? ...
6
votes
1answer
62 views

Is every nonzero integer the discriminant of some algebraic number field?

I do know that if $m \equiv 1 \mod 4$ and is squarefree, its probably the discriminant of $\mathbb{Q}[\sqrt{m}]$ and I also know some negative multiples of 27 are discriminants of cubic number ...
6
votes
2answers
126 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over ...
6
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2answers
230 views

How to compute a discriminant

Let $\alpha$ be a root of the irreducible cubic polynomial $x^{3}+px+q$, $p,q\in \mathbb{Q}$. How can I compute the discriminant $\Delta(1,\alpha,\alpha^{2})$ relative to $\mathbb{Q}(\alpha)$?
6
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1answer
279 views

Primes that ramify in a field

Consider the number field $L/\mathbb{Q}$. I know that the only primes $p$ that ramify over $L$ are the ones that divide $\Delta_{L}$, the discriminant of $L$. But what if I can't compute $\Delta_{L}$? ...
6
votes
3answers
218 views

When is a local algebra reduced?

Let $k$ be a field and let $A$ be a local $k$-algebra which has finite dimension over $k$. Let $\mathfrak{m}$ be the maximal ideal of $A$ and let $k' = A / \mathfrak{m}$ be the residue field. For ...
6
votes
1answer
286 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
6
votes
1answer
181 views

How to arrive at Ramanujan nested radical identity

I've come across a very curious Ramanujan identity $$\sqrt[6]{7 \sqrt[3]{20} - 19} = \sqrt[3]{\frac{5}{3}} - \sqrt[3]{\frac{2}{3}}$$ You could probably prove this by taking the 6th power of both ...
6
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1answer
153 views

Ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$.

I've seen that the ring of integers of $\mathbb{Q}(\sqrt{n})$ depends on $n\mod 4$. I am just wondering if we can (easily) write down the ring of integers of $\mathbb{Q}(\zeta_3,\sqrt[3]{2})$ (the ...
6
votes
1answer
167 views

What is the best book learn Galois Theory if I am planning to do number theory in future?

What is the best book learn Galois Theory if I am planning to do number theory in future? In a year i'll be joining for my Phd and my area of interest is number theory. So I want to know if there is ...
6
votes
1answer
227 views

Let $F_n$ be a Fibonacci number and $p$ a prime. Verify that for $p \le 61$, if $p\equiv\pm1 \pmod{5}$ then $p\mid F_{p-1}$

Define the Fibonacci entry point of $p$ to be the least integer $n$ such that $p\mid F_n$ So for example, for $p = 3$ - the Fibonacci entry point is $n = 4$ since $F_4 = 3$ and obviously $3\mid 3$. ...
6
votes
1answer
137 views

Are there Infinite Quotients of Algebraic Extensions of $\mathbb{Z}$?

It is well known that $\mathbb{Z}[a_1, \dots, a_n]/(a)$ is a finite ring if each $a_i$ is an algebraic integer and $a \neq 0.$ I suppose this statement becomes wrong if we just require those ...
6
votes
1answer
195 views

Why is this isomorphism $M \otimes_K L \stackrel{\simeq}{\longrightarrow} M^{[L:K]}$ an isomorphism of $M$ - algebras?

Suppose that $L/K$ is a finite separable extension of fields and let $M$ denote the Galois closure of $L$. Let $\textrm{Hom}_K(L,M)$ denote the set of all $K$ - algebra homomorphisms from $L$ to $M$. ...
6
votes
1answer
249 views

Finding the units in $\mathbb{Z}[\sqrt[3]{2}]$ and other questions

I'm reading a paper in which they solve the equation: $$a^3-2b^3=\pm 1$$ in integers using algebraic number theory. The number $a-b\alpha$, with $\alpha=\sqrt[3]{2}$, is a unit in ...
6
votes
2answers
281 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?
6
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3answers
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Proving Fermat's Last Theorem for n=3 using Euler's and Lamé's approach?

Euler and Lamé are said to have proven FLT for $n=3$ that is, they are believed to have shown that $x^3 + y^3 = z^3$ has no nonzero integer solutions. According to Kleiner they approached this by ...
6
votes
3answers
457 views

Is there any trivial reason $2$ is irreducible in $\mathbb{Z}[\omega],\omega=e^{\frac{2\pi i}{23}}$?

This naive question came as the last problem in my homework. The author asked me to use linear relations of the discriminant like ...
6
votes
1answer
650 views

Class number computation (cyclotomic field)

How does one prove that the class number of $\mathbb{Q}(\zeta_{23})$ is divisible by $3$? And afterwards how do you show that it is precisely $3$. Any help? Thanks in advance! //Ok, so I proved the ...
6
votes
2answers
474 views

On the class group of an imaginary quadratic number field

Let $d < 0$ be a square-free integer and let $p_{1},\ldots p_{r}$ be the prime divisors of $d$. Let $K := \mathbb{Q}[\sqrt{d}]$ and consider $P_{i} := (p_{i}, \sqrt{d}) \subset \mathcal{O}_{K}$. ...
6
votes
1answer
54 views

Haar Measure for Algebraic Number Theory: What Should I Know?

I recently taught myself some algebraic number theory and am preparing to take a course in class field theory this fall. I understand the notion of a Haar measure on a locally compact topological ...
6
votes
1answer
75 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
6
votes
1answer
264 views

Ring class field of $\mathbb{Q}(\sqrt{-19})$

I am looking an explicit form for the ring class field of the order $\mathbb{Z}[\sqrt{-19}]$ in the quadratic field $\mathbb{Q}(\sqrt{-19})$. Does anyone know if there is some and how it is?
6
votes
1answer
132 views

p-adic modular form example

In Serre's paper on $p$-adic modular forms, he gives the example (in the case of $p = 2,3,5$) of $\frac{1}{Q}$ and $\frac{1}{j}$ as $p$-adic modular forms, where $Q = E_4 = 1 + 540\sum ...
6
votes
1answer
115 views

Ramification indices and residue degrees of a finite Galois extension

Let $A$ be a dvr with fraction field $K$ of characteristic zero. Let $L/K$ be a finite Galois extension and let $B$ be the integral closure of $A$ in $L$. For a prime $b$ of $B$, let $e_b$ be its ...
6
votes
1answer
187 views

Factorization of primes and $Spec(\mathcal{O}_K)$

Let $K$ be a quadratic number field, and $\mathcal{O}_K$ the ring of integers of $K$. The map $\pi: Spec(\mathcal{O}_K) \rightarrow Spec(\mathbb{Z})$ that sends a prime ideal $\mathbb{p}$ to ...
6
votes
1answer
786 views

Ramification index and inertia degree

Let $L,K$ be number fields and $L|K$ a galois extension. Let $(0)\neq Q$ a prime ideal in $\mathcal O_L$ (=ring of integers in $L$) and $P=Q \cap \mathcal O_K$. $Z_Q $ denotes the decomposition ...
6
votes
2answers
73 views

Checking irreducibility of polynomials over number fields

Are there general methods for checking irreducibility of polynomials over number fields? For instance, letting $F = \mathbb{Q}(\sqrt{3})$, I want to know whether $x^3 - 10 + 6\sqrt{3}$ is irreducible ...
6
votes
1answer
123 views

Eigenvalues of Multiplication in algebraic number field

Finally I have a new question which is puzzling me. I am currently reading a manuscript so there is no reference or link I can provide. I will try to put all the information needed in here. This ...
6
votes
1answer
107 views

Factorization of $5$ in the splitting field of $x^3 + 2$

I wonder if someone could help to clarify the following. Let $\zeta$ be a primitive cube root of unity and $\alpha = \sqrt[3]{2}$. Let $K = \mathbb{Q}(\alpha)$ and $L = K(\zeta)$. Then $L$ is the ...
6
votes
1answer
378 views

Positivity of the norm of an element of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be an $l$-th primitive root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field and $\alpha$ be a non-zero element of ...