Questions related to the algebraic structure of algebraic integers

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5
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1answer
68 views

Importance of continuity of Galois representations

So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is ...
1
vote
1answer
78 views

The order of the cokernel of an endomorphism over $ \mathbb Z_p$

I want to prove the following result : Let $X$ a finite-rank free $\mathbb{Z}_p$-module, and $\varphi \colon X \to X$ an endomorphism of $X$. Then $$|M/\varphi(X)| < \infty \Leftrightarrow ...
6
votes
1answer
112 views

Equivalent Definitions of Ideal Norm

Let $A \subseteq B$ be Dedekind domains, with $K \subseteq L$ their quotient fields with $L/K$ finite and separable. If $J$ is a fractional ideal of $B$, then $$ J = \prod\limits_{\mathfrak B} ...
5
votes
1answer
101 views

How should I think about Ihara's Lemma?

I am trying to learn about Ihara's lemma because I see it mentioned in many papers in arithmetic geometry. To be more precise, I would like to know: What is the significance of this result? Why is ...
1
vote
2answers
46 views

Suppose $γ$ is a $k$th root of unity that satisfies a quadratic equation $z^2−mz−n=0$ with $m,n\in\mathbb{Z}$. Then $k=3,4$ or $6$

Let $k\in\mathbb{Z}$ with $k>2$ and suppose $\gamma$ is a $k$th root of unity that satisfies a quadratic equation $z^2-mz-n=0$ with $m,n\in\mathbb{Z}$.Then $k=3,4$ or $6$. My knowledge on ...
3
votes
0answers
78 views

Minimal polynomial of eigenvector entries

Suppose that $M$ is a matrix with integer entries and that $\lambda$, $v = (v_1, \ldots, v_n)^{T}$ are an eigenvalue and eigenvector of $M$. Then $\lambda$ is an algebraic number and we can see this ...
6
votes
0answers
208 views

Generalization of a result on finite Galois extensions to infinite case.

It is well known that if $L/K$ is a finite abelian extension, $\frak p$ a prime of $K$ and $\frak P$ a prime of $L$ above $\frak p,$ then $L^I /K$ is the maximal subextension of $L/K$ in which ...
3
votes
2answers
107 views

Primes of the form $x^2 + ny^2$: necessary and sufficient condition not in terms of a “modulo equation”?

For particular cases ($n=1,2,3$) we can find an "elementary" necessary and sufficient condition, i.e. $$p = x^2 + y^2 \Leftrightarrow p \equiv 1 \mod 4$$ $$p = x^2 + 2y^2 \Leftrightarrow p \equiv 1,3 ...
0
votes
0answers
55 views

Conditions on $a,b\in\mathbb{Q}$, for $a+b\sqrt{n}$ to be integral over $\mathbb{Z}$

For $n\in \mathbb{Z}$ square-free, let $$k:=\mathbb{Q}(\sqrt{n}),$$ and let $$\alpha:=a+b \sqrt{n}\in k.$$ Prove that $$ \alpha \mbox{ is integral over } \mathbb{Z}\;\;\; \Longleftrightarrow ...
5
votes
2answers
70 views

Elements of order 2 in the absolute Galois group

So I remember reading once that the only element of $G=Gal(\overline{\Bbb Q} / \Bbb Q)$ that we understand is complex conjugation. Suppose we fix an embedding of $\overline{\Bbb Q}$ into $\Bbb C$. ...
4
votes
5answers
149 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
3
votes
2answers
86 views

Places ramifying in an extension of number fields

I came across the following statement in a number theory paper: Let $L/K$ denote an arbitrary Galois extension of number fields with Galois group $G$. Let $S$ be a finite non-empty set of places of ...
1
vote
1answer
64 views

Prime Splits Completely in Every Intermediate Field

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits ...
6
votes
0answers
141 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
1
vote
1answer
100 views

Pro$-p-$group as a $\Lambda-$module

Let $p$ be a prime number, and let $X$ be an abelian pro$-p-$group (i.e for some indexing set $I,$ we have $X=\varprojlim X_i$ where $X_i$ is a finite, abelian $p-$group for each index $i \in I .$) ...
3
votes
2answers
52 views

Unique factorization of Ideals?

Is it case that even if the domain is not UFD for its elements, the domain is UFD for ideals. I mean can we uniquely factorized the ideals, whatsoever? possible, and why? for example, in ...
0
votes
1answer
48 views

If $\frac{a+b\alpha+c\alpha^2}{3} \in R$, then $a\equiv b \equiv c \equiv 0 \pmod3$

If $\frac{a+b\alpha+c\alpha^2}{3} \in R$, then $a\equiv b \equiv c \equiv 0 \pmod3$ where $R=\mathcal{O}\cap\mathbb{Q}[\alpha]$ I know that if $\frac{a+b\alpha}{3}$, then $a\equiv b \equiv 0 ...
1
vote
0answers
52 views

Topological isomorphism of projective system

Let $K_0 \subset K_1 \subset ...K_n\subset...$ be a tower of normal number fields, and put $G_n = \mathrm {Gal} (K_n/K_0).$ Define epimorphisms $\pi_{mn} :G_n\longrightarrow G_{m}$ for $m < n$ ...
2
votes
1answer
84 views

Non-unique factorization of an ideal in UFD

In Z[x], the ideal <2, x> is not principal. I am that the factorization of a nontrivial ideal into prime ideals is unique in a Dedekind domain. Not all UFD are Dedekind domain, so there must be a ...
0
votes
2answers
124 views

What's a Dual Basis?

If $L/K$ is a finite extension of fields with $v_1, ... , v_n$ a basis, then we have an isomorphism of $K$-modules $L/K \rightarrow Hom_K(L,K)$, where a basis for $Hom_K(L,K)$ is the "dual basis" of ...
2
votes
1answer
54 views

Projective coordinates for elliptic curves

If we consider an elliptic curve projectively, it is a homogeneous form in $3$ variables say $x$, $y$ and $z$. How is this related to the Thue equations (homogeneous forms in $2$ variables)? I'm ...
0
votes
1answer
36 views

if $\beta\in \mathcal{O}\cap\mathbb{Q}[\mathbb{w}]$, then $\beta=a_{0}+a_1w+…+a_{p-1}w^{p-1}$

$w=e^\frac{2i}{p}$ where p is odd prime. $\mathbb{Z}[]$ if $\beta\in \mathcal{O}\cap\mathbb{Q}[\mathbb{w}]$, then $\beta=a_{0},+a_1w+...+a_{p-1}w^{p-1}$ where $a_i's$ are unique integers this a ...
2
votes
1answer
91 views

How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}[\alpha]$

How to show that $1,\alpha,\alpha^2/2$ is an integral basis of $R=\mathcal{O}\cap \mathbb{Q}[\alpha]$. ($\mathcal{O}$ is ring of algebraic integers) $\alpha$ is a root of $f(x)=x^3+2x^2+4$ which is ...
2
votes
1answer
42 views

How to show $a^2+2B^2=p$ has integer solutions for all primes p with $(\frac{−2}{p})=1$

How to show $a^2+2b^2$=p has integer solutions for all primes p with $(\frac{−2}{p})=1$ (legendre symbol) Partial solution: $(\frac{−2}{p})=1$ $\Rightarrow$ p $\ |$ ...
2
votes
2answers
122 views

AIME number theory problem (unique factorization domains)

I'd greatly appreciate some help with the following problem, from a mock AIME I took. Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
-2
votes
1answer
93 views

Compute the class number of $R=\mathcal{O}\cap\mathbb{Q}[\sqrt{51}]$

What is the class number of $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{51})$. Could you please explain.
0
votes
2answers
52 views

How to show that either $a^2+5b^2=p$ or $c^2+5d^2=2p$ has integer solutions for all prime $p$ with $(\frac{-5}{p})=1$

How to show that either $a^2+5b^2=p$ or $c^2+5d^2=2p$ has integer solutions for all prime $p$ with $(\frac{-5}{p})=1$ would the fact that $\mathbb{Z}[\sqrt{-5}]$=$\mathcal{O}\cap ...
5
votes
4answers
260 views

Give an example of a UFD having a subring which is not a UFD.

Give an example of a UFD having a subring which is not a UFD. I thought of $\mathbb{Z}[\sqrt{2},\sqrt{3}]$. Could you please explain my question. I am trying grasp the concepts, need help.
0
votes
1answer
46 views

How to show that every prime ideal 0f R contains a unique prime of $\mathbb{Z}$

Let $K$ be a number field with the ring of integers of $R$. How to show that every prime ideal of $R$ contains a unique prime of $\mathbb{Z}$. I have no idea, could you please help.
1
vote
1answer
35 views

Fundamental unit

Let K is a cubic extension of $\Bbb{Q}$ having only one real embedding in $\Bbb{R}$, then can we find fundamental unit in the ring of integers which is real.
4
votes
1answer
71 views

Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...
1
vote
0answers
59 views

Question on units in a cubic number field

If $K$ is a cubic extension of $\Bbb Q$ (the rational numbers) having only one real embedding in $\Bbb R$ ( the real numbers), then why should the units in the ring of integers be of the form $\pm ...
2
votes
1answer
101 views

Find the ideal class group of $\mathbb{Z}(\frac{1+\sqrt{-31}}{2})$.

Find the ideal class group of $\mathbb{Z}(\frac{1+\sqrt{-31}}{2})$. I found the the ideal class group(ICG) is generated by primes ideals lying over 2 and 3. $\lt 2 \gt$ =$P_2 \hat P_2$ $\lt 3 \gt$ ...
2
votes
2answers
78 views

Find all ideals in $\mathcal{A}\cap \mathbb{Q}[\sqrt{-5}]=\mathbb{Z}[\sqrt{-5}]$ that contain 30.

Find all ideals in $\mathcal{A}\cap \mathbb{Q}[\sqrt{-5}]=\mathbb{Z}[\sqrt{-5}]$ that contain 30. so far, we found, $\lt 2 \gt$ =$\lt 2, \alpha+1\gt^2$=$P^2_2$ $\lt 3 \gt$ = $\lt 3,\alpha - 1\gt ...
1
vote
0answers
32 views

residual degree in $\mathbb{Q}(i,\sqrt[4]{2})$

Let $K = \mathbb{Q}(i,\sqrt[4]{2})$ (i.e. the decomposition field of $X^4-2$ over $\mathbb{Q}$). I want to compute the residual degree of $3$ in $K$. Here is how I proceeded : We know that $3$ is ...
1
vote
3answers
67 views

Divisors of $mn$ in Rings without Unique Prime Factorization

Using the fundamental theorem of arithmetic, it's easy to prove this proposition: Proposition. Every divisor of $mn$ can be written as the product of a divisor of $m$ to a divisor of $n$. My ...
5
votes
2answers
49 views

Norm of $\mathbb{Q}_2(i)^\times$

I am trying to compute the norm group of $\mathbb{Q}_2(i)^\times$. If i'm not mistaken we have $\mathbb{Q}_2(i)^\times = (1+i)^\mathbb{Z}\mathbb{Z}_2[i]^\times$ so $N(\mathbb{Q}_2(i)^\times) = ...
5
votes
2answers
128 views

$||x||=1$ in $K/\mathbb{Q}$ implies $x$ is a root of unity.

Let $K/\mathbb{Q}$ a finite (i.e. algebraic and finitely generated) extension. Let $x \in K$, such that $||x||=1$ for all normalized absolute values of $K$ but at most one. Then $x$ is a root of ...
24
votes
5answers
401 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
0
votes
0answers
23 views

Gauß sum and conductor $|\tau(\chi)|^2 = f_\chi$

From Marcus "Number Fields" on page 200: I am working through the proof there, but I don't understand one step. First the proof: $|\tau(\chi)|^2 = \tau(\chi)\bar{\tau(\chi)} = ...
6
votes
2answers
117 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 ...
0
votes
1answer
82 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
0
votes
1answer
27 views

Gauß sum and primitive character

I am working with Daniel Marcus "Number Field" Book. And I have a question to the following Lemma: $$\tau_k(\chi)=\left\{\begin{array}{ll} \bar\chi(k)\tau(\chi), & \textrm{if }(k,m)=1 \\ ...
6
votes
2answers
62 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 ...
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votes
1answer
60 views

Sequence of numbers with a special property [closed]

Prove that the sequence a(n) = 2013 + 317n, where n is any nonnegative integer, generates infinitely many palindromic numbers.
3
votes
1answer
88 views

Introductive Book on Modular Forms

I'm looking for an introductive book on Modular Forms and their applications to Algebraic Geometry and Algebraic Number Theory. Some Ideas? Explaining you my prerequisites, I've a good knowledge of ...
8
votes
0answers
312 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
0
votes
0answers
42 views

Notation: $p^*$ for a prime $p$

I'm reading a paper which uses $p^*$ for some prime $p$, which is then defined as $(-1)^{(p-1)/2} p$, but I can't seem to get a clear distinction between where $p$ and $p^*$ are used. The paper also ...
0
votes
0answers
28 views

Question regarding solution of a problem in Problems in Algebraic Number Theory?

I believe either the question or the solution are wrong, but I may be missing something. The question for Exercise 6.5.24 of Problems in Algebraic Number Theory (2nd edition) by M. R. Murty and J. ...
1
vote
1answer
45 views

nth root of unity in $\mathbb{Q}_p $ [closed]

Let $n\in \mathbb N,$ and $p$ be a prime number. Let $\zeta_n$ be a nth root of unity in $\overline{\mathbb Q_p}.$ Under what conditions we have $\zeta_n\in\mathbb Q_p$ ?