Questions related to the algebraic structure of algebraic integers

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5
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1answer
155 views

Serre's Modularity Conjecture — Weight

I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ...
1
vote
0answers
20 views

Why is the norm of an ideal contained in that ideal?

Suppose $K$ is a number field and that $\mathcal{O}_K$ is the ring of integers of $K$. Now, let $I$ be an ideal in $\mathcal{O}_K$. I know that $N(I) \in I$, but I want to prove it. By definition, ...
0
votes
3answers
66 views

Prove some number is algebraic over a field

How do you prove (without calculating the minimum polynomial) that $\sqrt{3}$ + $\sqrt[]{5}$ is algebraic over $\mathbb{Q}$. Also prove that $\left(\mathbb{Q}(\sqrt{3} + \sqrt[]{5}\right):\mathbb{Q}) ...
4
votes
2answers
112 views

The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.

I have to study the prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
6
votes
2answers
65 views

Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
3
votes
1answer
29 views

Associated order of a Galois extension of number fields

I have a finite Galois extension $L/K$ of number fields with group $G$. Let $\mathcal O_L$ be the ring of algebraic integers of $L$. We let $\mathcal A_{L/K}$ be the subring $\{x\in K[G]:x\mathcal ...
0
votes
1answer
42 views

Polynomial Diophantine Equations

So in general how does one decide if: $$ a_0 + a_1x + a_2x^2 ... a_{n_1}x^{n_1} = b_1y + b_2y^2 ... + b_{n_2}y^{n_k}$$ Has solutions for integers $x,y$ given real numbers $a_0, a_1. .. a_{n_1}, b_1 , ...
6
votes
1answer
83 views

Is there a simple procedure to produce algebraic numbers of modulus one that are not roots of unity?

Let $z=e^{i\theta}$ be a complex number of modulus $1$. Trivially if $z$ is a root of unity then $z$ is also an algebraic number, but the converse is known to be false : $z$ can be algebraic without ...
1
vote
2answers
51 views

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y [duplicate]

Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y ...
6
votes
2answers
54 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 ...
9
votes
1answer
240 views

Why is $(\sqrt{2}+\sqrt{3})^{2008}$ so close to an integer?

Using 5000-digit precision in PARI/GP, I discovered that the fractional part of $(\sqrt{2}+\sqrt{3})^{2008}$ is extremely small, less than $10^{-999}$. Is there a simple explanation for this fact ? ...
2
votes
1answer
37 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
3
votes
1answer
28 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
0
votes
0answers
40 views

divisibility question involving primes

I have a question concerning the following divisibility problem. For any prime $p$ we define set: $\mathtt{V_{p}}:=\Biggl\{F\in\Phi\Biggl|\begin{cases}p^2\nmid ...
0
votes
1answer
51 views

Before we consider the prime decomposition

Let $L/K$ be a number field extension. Let $I$ be a prime ideal of $O_K$. How to prove that $IO_L\neq O_L$? It looks there should be a very fast way to see this, but I don't know how.
2
votes
2answers
61 views

Transcendentals as the Roots of Infinite Polynomials

I have always been taught that the difference between an algebraic and a transcendental number is that the former is the root to a polynomial of ${\bf finite}$ degree with integer coefficients. I did ...
4
votes
1answer
57 views

Find the prime decomposition of $(2) $ in $\mathbb Z\left[ \frac{ 1 + \sqrt n }{2 } \right]$

Let $n$ be an integer such that $ n \equiv 1 $ mod $4$. Let $\mathbb Z \left[ \frac{ 1 + \sqrt n }{ 2} \right]$ be our ring. Let $(2)$ be the ideal generated by $2$. What is the prime ...
0
votes
0answers
27 views

Discriminant remains unchanged when reducing mod p

Let $ \theta $ be an algebraic integer and let $g(t) \in \mathbb Z [ t ] $ be its minimal polynomial over $\mathbb Q$. Let $ \bar g (t) \in \mathbb F_p [ t] $ be the same polynomial with coefficients ...
1
vote
1answer
41 views

Show that a polynomial does not have multiple roots

Let $K$ be a number field and $\theta $ be an algebraic integer such that $ K = \mathbb Q(\theta )$. Let $ d $ be the discriminant of the basis $\{ 1, \theta, ..., \theta ^{ n -1 } \}$ of $K$ over ...
6
votes
1answer
93 views

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

I do know that if $m \equiv 1 \pmod 4$ and squarefree, it is probably the discriminant of $\mathbb{Q}[\sqrt{m}]$, and I also know some negative multiples of 27 are discriminants of cubic number ...
5
votes
1answer
93 views

Gentle introduction to algebraic number theory

For context, I am a undergraduate majoring in math. I've taken two semesters of algebra (though I am still a bit shaky on Galois theory). I just finshed a course in elementary number theory which used ...
9
votes
5answers
221 views

Are numbers of the form $n^2+n+17$ always prime

Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it ...
0
votes
1answer
34 views

Prove that $(1,\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{13}}{2},\frac{1+\sqrt{5}+\sqrt{13}+\sqrt{65}}{4})$ is an integral basis for $K$

Prove that $\left(1,\frac{1+\sqrt{5}}{2},\frac{1+\sqrt{13}}{2},\frac{1+\sqrt{5}+\sqrt{13}+\sqrt{65}}{4}\right)$ is an integral basis for $K=\Bbb{Q}(\sqrt{5},\sqrt{13})$. What is $d(K)$? I ...
1
vote
1answer
31 views

Norms of Ideals and generators.

I'm self studying some Algebraic Number Theory, looking at norms of ideals within rings of integers for some number field. I know that if we have a principal ideal $I= (a)$, then the norm of the ...
0
votes
1answer
43 views

Quadratic residue modulo $p$ iff quadratic residue module $p^k$

Let $p$ be an odd prime, $a\in \mathbb{Z}$ with $(a,p)=1$. I am trying to show that if $a$ is a square modulo $p$ then it is a square modulo $p^k$. I managed to prove this using an exponential ...
3
votes
1answer
52 views

How do I prove $2$ is prime (or irreducible) in all $Z[\sqrt{d}]$ with $d < -7$?

How do I prove $2$ is prime (or irreducible) in all $Z[\sqrt{d}]$ with $d < -7$? I feel like the answer is right under my nose, but I just can't see it.
2
votes
1answer
33 views

Evaluating $j$-invariant in PARI/GP.

Is there any command to evaluate $j$-invariant in PARI/GP? In Pari/Gp reference card there is $\operatorname{ellj}(x)$ function; but I am not understanding how to evaluate $j(i)$ or ...
1
vote
4answers
168 views

Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$

As suggested on Mathoverflow (http://mathoverflow.net/questions/168536/solutions-to-the-diophantine-equation-2a2-2b2-c2-d2-0) I am transfering this question to math-stackexchange: I am looking for ...
0
votes
1answer
54 views

Show that this is a finitely generated abelian group and compute its rank

If $n$ is a square-free integer such that $n >1$, and $ K = \mathbb Q ( \sqrt n )$. Let $ A_K$ the ring of algebraic integers. Show that $ A_K ^ \times $ is a finitely generated abelian group and ...
2
votes
1answer
35 views

Ideals in a Quadratic Number Field

Show the ideal $I=\langle4,2+2\sqrt{-29}\rangle$ in $\mathbb{Z}[\sqrt{-29}]$ satisfies the equality $\langle8\rangle=I^{2}$ of ideals in $\mathbb{Z}[\sqrt{-29}]$. I tried to factorise $x^{2}+29$ over ...
3
votes
1answer
36 views

a step in a proof in Samuel's Algebraic number theory

In the proof of Dirichlet's unit theorem, in Algebraic number theory by Samuel, there is a step in the proof that i don't understand. (p.73 in the french edition). He first introduces the logarithmic ...
1
vote
2answers
44 views

Algebraic number raised to a potence

Is that true that an arbitrary albegraic number raised to a potence is again algebraic? That´s false, there are many examples to found. Now, is that true that an algebraic number raised to a rational ...
2
votes
1answer
41 views

Norm map of totally ramified extension

There is a nice characterization of the image of the norm map of $F$ over $K$, where $F$ is an unramified extension over $K$. More precisely, $N_{F/K}(F^{\times})=\lbrace u \pi^n| u\in ...
0
votes
3answers
118 views

Calculating the class group of $\mathcal{O}_K$, for $K=\mathbb{Q}(\sqrt{7})$?

How to calculate the class group of $\mathcal{O}_K$, for $K=\mathbb{Q}(\sqrt{7})$ without using the Minkowski bound?
3
votes
1answer
62 views

How to prove a number is algebraic? [closed]

Show that $\sqrt[3]{\sqrt 3+2} +4$ is an algebraic number
6
votes
1answer
127 views

Are there relations among Frobenii?

Let $G=\text{Gal}(\overline{\mathbf Q}/\mathbf Q)$, and for each prime $p$, choose an embedding $\overline{\mathbf Q} \hookrightarrow \overline{\mathbf Q_p}$. Let $\sigma_p$ be a choice of Frobenius ...
1
vote
0answers
50 views

Algebraic extension of an abelian group.

Let $H \leqslant G$ be abelian groups. Suppose there were $k \gt 1$, $x \in G \setminus H$, such that $x^k = b \in H$. Then Define $H(x) = \{h x^n : n \in \Bbb{Z}, h \in H\}$ to be a simple ...
4
votes
1answer
91 views

Question on extensions of discrete valuation fields

Let $F$ be a discrete valuation field. Let $L$ be a finite extension of $F$. Let $L=F(\alpha)$ where $\alpha$ belongs to ring of integers of $L$, denoted by $O_L$. Is it always true that ...
3
votes
0answers
23 views

what is the unique prime factorization for the ideal $p\mathbb{Z}[\zeta]$ in the Dedekind domain $\mathbb{Z}[\zeta]$?

Let $p$ be a prime number and $\zeta=e^{\frac{2\pi}{p}i}$. I want to find the unique factorization into a product of prime $\mathbb{Z}[\zeta]$-ideals for the ideal $p\mathbb{Z}[\zeta]$. Now, I know ...
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0answers
61 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
2
votes
0answers
35 views

$\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$

prove that $\{1,\sqrt[3]{3},(\sqrt[3]{3})^2 \}$ is integral basis for $\Bbb{Q}(\sqrt{3^3})$ my aim is to show that $\Bbb{Q}_K=\Bbb{Z}+\sqrt[3]{3}\Bbb{Z}+(\sqrt[3]{3})^2\Bbb{Z}$ $\supseteq$ : ...
3
votes
2answers
60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
2
votes
2answers
127 views

Silly mistake in this number theory book

My question is very easily to be solved (at least I hope so) I think this book has a mistake: When I calculate I get $b_3\equiv -2 (\mod{2})$ which implies $b_3=0$, am I right? Another question, ...
0
votes
1answer
59 views

Is this ring an integral domain?

I'm starting to study Algebraic number theory and I'm having problems with the first examples of this book. I'm trying to prove this is a quadratic domain, i.e., an integral domain: I'm sorry I ...
1
vote
1answer
56 views

Integral Closure in $\mathbb{Q}(\alpha)$

Let $\alpha $ be a root of $f(x) = x^3 -x +3$, $K=\mathbb{Q}(\alpha)$ I have to prove $\mathcal{O} _K = \mathbb{Z}[\alpha]$ for the integral closure $\mathcal{O} _K$. I have shown that the disciminant ...
1
vote
0answers
26 views

$\zeta$ primitive $n$th root of unity, help showing that $\sqrt{n},\sqrt{-n}\in \mathbb{Q}(\zeta)$ under some conditions.

Consider $\zeta$ a primitive $n$th root of unity, show that if $n\equiv 1\mod{4}\implies \sqrt{n}\in \mathbb{Q}(\zeta)$ if $n\equiv -1\mod{4}\implies \sqrt{-n}\in \mathbb{Q}(\zeta)$. I know that ...
2
votes
0answers
33 views

Dedekind's criterion clarification

Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf) Is it true that the prime ideals ...
12
votes
1answer
127 views

When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that ...
1
vote
1answer
27 views

Normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ is UFD…

We know that the normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ where $d\in \mathbb{Z}$ is $$O=\mathbb{Z}[\beta], \text{$\beta=\sqrt{d}$ if $d\equiv2,3 \pmod 4$}; \ \frac{1+\sqrt{d}}{2} ...
6
votes
1answer
78 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. ...