Tagged Questions

Questions related to the algebraic structure of algebraic integers

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3
votes
1answer
86 views

can anyone give a proof by definition :11 is prime in $ \mathbb{Z}[\sqrt{-5}] $

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $ \alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma) $ then ...
0
votes
2answers
28 views

Norm in algebraic number fields

Consider an algebraic number field $\mathbb{Q}(\alpha)$ and it's ring of integers $O$. If we take any element $\xi \in O$ and we want to calculate it's norm $N_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\xi)$, ...
3
votes
1answer
35 views

ramification index of $p$ in $\mathbb{Z}\left[e^{\frac{2\pi i}{p}}\right]$

I am attempting to show that $p$ has ramification index $p-1$ in $\mathbb{Z}[\omega]$ where $\omega=e^{2\pi i/p}$. The issue is I want to do so avoiding actually factoring $p$. I was hoping to use ...
3
votes
1answer
22 views

Help unmasking a disguised principal ideal

I recently saw a question on here about trying to generate a non-principal ideal in a principal ideal domain, with the only answer so far saying that if the ring $R$ is a PID, then $\langle e, f ...
2
votes
1answer
36 views

Algebraic number field with non trivial integral basis

So far I have only seen extensions of $\mathbb{Q}$ with "trivial" integral basis. Meaning that the integral basis is the most natural one e.g. the integral basis for $\mathbb{Q}(\sqrt[3]{2})$ is just ...
1
vote
1answer
49 views

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$. I'm trying this problem with the Minkowski bound, please I need more help. Thanks
3
votes
1answer
34 views

In a PID, does every attempt to generate a non-principal ideal just lead back to the whole ring itself?

It is a well-known fact that a unique factorization domain is a principal ideal domain, in which all ideals are principal ideals. [EDIT: I got dyslexic on this one, should've said something along the ...
0
votes
1answer
30 views

Field, algebraic element

1) Let $E/F$ an extension and let $\alpha,\beta\in E$ be algebraic elements over $F$. If $\alpha\neq 0$, prove that $\alpha+\beta$, $\alpha\beta$ and $\alpha^{-1}$ are all algebraic over $F$. 2) If ...
2
votes
1answer
47 views

Show that $x$, $y$, $z$ are integers when $3x$, $3x^2-6yz$, $x^3+2y^3+4z^3-6xyz$ are integers.

I was trying to show that $\{1, \alpha, \alpha^2\}$ is a integral basis of $\mathbb{Q}(\alpha)$ where $\alpha= \sqrt[3]{2}$. And after some steps it remains to prove that if $$3x, \quad 3x^2-6yz, ...
2
votes
0answers
42 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
1
vote
0answers
68 views

Lattices in $\mathbb C$ as modules of the ring of integers in an imaginary quadratic field

Let $K$ be an imaginary quadratic number field and let $O_K\subset K$ be the ring of algebraic integers in $K$. Let us call a lattice $\Lambda\subset\mathbb C$ normalized if the tori $\mathbb ...
3
votes
1answer
41 views

What is the class number of $\mathcal{O}_{\sqrt[3]{18}}$?

I accept it without proof that $\mathcal{O}_{\sqrt[3]{2}}$ and $\mathcal{O}_{\sqrt[3]{3}}$ both have class number $1$. Also, I've been told that $\mathcal{O}_{\sqrt[3]{m^2}} = ...
1
vote
0answers
35 views

What is the definition of the completion of a $\mathfrak o_K$-module at an infinite prime?

I have a number field $F$ and its ring of integers $\mathfrak o$ and an infinite place $\mathfrak p$ of $F$. Let $V$ be a finitely generated $\mathfrak o$-module. My question is, how is the ...
4
votes
1answer
75 views

$\sum_{\zeta^p=1}(\zeta-1)^n$

Given $n\geq0$ let $$ z_n=\sum_{\zeta^p=1}(\zeta-1)^n $$ where $p$ is an odd prime number (summation extended to all $p$-th roots of 1). It is clear that: $z_n\in\Bbb Z$ (it's a Galois invariant sum ...
0
votes
1answer
28 views

Completion of a number field at a complex embedding

Sorry if this question has been asked before. Let $K$ be a number field of degree $n>1$ and $\sigma:K\hookrightarrow \mathbb C$ a complex (non real) embedding of $K$ in $\mathbb C$ giving the ...
2
votes
0answers
28 views

Identity relating Frobenius to multiplication by p on the ring of Witt vectors

Let $k$ be a field of characteristic $p$, $F$ be the Frobenius map of Witt vectors, and $V$ the transfer map on $W(k)$. I'm trying to show that $FV(a) = pa$ where $a$ is a Witt vector. Clearly this is ...
6
votes
4answers
87 views

Uniqueness of representation of prime as $x^2+2y^2$

It can be proven that every prime $p\equiv1,3\mod{8}$ can be written in the form $a^2+2b^2$. Is it true that this representation is unique? This is certainly true for primes written in the form ...
3
votes
1answer
28 views

Density of ring of algebraic integers in $\mathbb C$

Clearly, $\mathbb Z$ is not dense in $\mathbb Q$ (and not dense in $\mathbb C$). But why is the ring of algebraic integers $\overline{\mathbb Z}$ dense in $\overline{\mathbb Q}$? In particular, if ...
1
vote
0answers
32 views

Exercise 2.8 Cassels and Frohlich

I don't understand the discussion in exercise 2.8 of Cassels and Frohlich (page 352) beginning with "more generally". Why should it matter whether the formula for $c$ has a power of $-1$ in it if this ...
3
votes
0answers
30 views

Class number 1 for negative integers

If n is a positive integer then, $h(-4n) = 1 \iff n = 1,2,3,4,7$. The only proof I have seen of this is long and case wise. Is there any conceptual a priori reason to believe that these numbers are ...
0
votes
1answer
40 views

How can I see that $4$ is not a quartic residue?

How can I see that $4$ is no quartic residue, i.e. there is no $t$ such that $t^4 \equiv 4 \mod p$ when $p\equiv 5 \mod 8$?
3
votes
1answer
49 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some ...
3
votes
1answer
23 views

Extending a DVR could produce not a DVR

I'm reading Tate's paper about $p$-divisible groups. In Chapter $(2.4)$ he asserts that if you take $R$ a complete DVR with residue field $k$ of characteristic $p>0$, $K$ its field of fractions, ...
9
votes
0answers
81 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
0
votes
1answer
22 views

Is it true that an equivalent 'absolute value' is an absolute value?

I've a very basic question on absolute values on fields. If $K$ is a valued field with absolute value $|- |:K\to \mathbb R_{\geq0}$ then is the map $|-|':K\to \mathbb R_{\geq0}$ defined by ...
4
votes
3answers
97 views

Show that $\sqrt [3]{2}-\sqrt [3]{4}$ is algebraic

How do I show, step by step, that $\sqrt [3]{2}-\sqrt [3]{4}$ is a root of $x^3+6x+2$? Start with $x=\sqrt [3]{2}-\sqrt [3]{4}$ do not use the cubic, the cubic is given for convenience. ( This is ...
0
votes
1answer
12 views

the number of non-zero integral ideals of norm m in a ring of integers [closed]

How to prove that the number of non-zero integral ideals of norm m in a ring of integers of a number field with degree n is less than or equal to the number of n-dim vectors of n positive integer ...
0
votes
0answers
61 views

An Efficient Route to Tate's Thesis

I want to learn Tate's thesis. My advisor suggested the Book "Algebraic Number Theory" By Lang. However, it seems to be a long read before I reach Tate's Thesis. I want to know what are other good ...
3
votes
1answer
46 views

Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
1
vote
1answer
42 views

Find the Value of $n$ Where $15756$ is the $nth$ Member of A Set

It's a question from $BNMO$.It still haunts me a lot. I want to find an answer to this question. Any number of the different powers of $5: 1,5,25,125$ etc is added one at a time to generate the ...
1
vote
0answers
21 views

Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
0
votes
0answers
17 views

Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that ...
3
votes
1answer
29 views

valuation ring, completeness

Perhaps a trivial question: is there an example of a field $K$ and a valuation $v$ on $K$ such that the following holds: $K$ is not complete (with respect to the valuation topology) The valuation ...
2
votes
1answer
99 views

$x^{16}-16 \equiv 0 \mod p$ has a solution for each prime

I have to prove that $x^{16}-16\equiv 0 \mod p$ has a solution for every prime $p$. I already know (from a previous work) that $x^8-16\equiv 0 \mod p$ has a solution for every prime. In my opinion, I ...
4
votes
1answer
35 views

sequence $\{a^{p^{n}}\}$ converges in the p-adic numbers.

Let $a\in \mathbb{Z}$ be relatively prime to $p$ prime. Then show that the seqeunce $\{a^{p^{n}}\}$ converges in the $p$-adic numbers. This to me seems very counter intuitive. Since $(a,p)=1$ the ...
1
vote
0answers
38 views

Topology on ideles

I am new to adeles and ideles in number fields. I am trying to solve Exercise 2 in this pdf. This is a very standard fact the statement of which can be found in any textbook. I have done exercise 2 ...
2
votes
1answer
58 views

Confusion on Inert Primes in Ireland and Rosen

In Ireland and Rosen, the following law for inert rational primes in a quadratic field is stated as: if $p\nmid \delta_K$, where $\delta_K$ is the discriminant of the quadratic field, and $d$ is a ...
3
votes
1answer
41 views

Kummer-Dedekind's factorisation theorem

For a number field extension $K$ of $\mathbb{Q}$ one can factorise almost all prime ideals $(p)$ in the extension $K$, except finitely many, easily by factorising minimal polynomials in finite ...
1
vote
2answers
64 views

What does $x$ $\in$ $\mathbb{Q}$(y) mean?

What does $x$ $\in$ field $\mathbb{Q}$(y) mean? What is $\mathbb{Q}$(10), for instance?
1
vote
0answers
19 views

Why do we conclude that $(a,b)=1$, having found that $(a',b'=1)$?

Suppose that we have the equation $ax^2+by^2+cz^2=0, a,b,c \in \mathbb{Q}$. Without loss of generality, we suppose that $gcd(a,b,c)=1$. Also, we can consider that $a,b,c$ are square-free. We can ...
1
vote
1answer
30 views

convergence of the sequence $10^{-n}$ in the p-adic numbers

Let $p$ be prime. I am tasked to prove that the sequence $10^{-n}$ does not converge in $\mathbb{Q}_{p}$ for any $p$ where $\mathbb{Q}_{p}$ is the set of p-adic numbers. For $p=2$ or $5$, we see ...
0
votes
2answers
58 views

Example of non fractional ideal

Let $R$ be an integral domain with fraction field $K$, and let $I$ be an $R$-submodule of $K$. We say that $I$ is a fractional ideal of $R$ if $rI\subset R$ for some nonzero $r \in R$. My question ...
3
votes
2answers
88 views

Show that $\mathbb{Z}[\sqrt{223}]$ has three ideal classes.

Well the question is the title. I tried to grab at some straws and computed the Minkowski bound. I found 19,01... It gives me 8 primes to look at. I get $2R = (2, 1 + \sqrt{223})^2 = P_{2}^{2}$ $3R ...
0
votes
0answers
54 views

How to find discriminant of a given polynomial.

How to find discriminant of the polynomial- $x^7 - x^5 + x^3 - x + 1$. Discriminant of a polynomial is given by- $D(f)=\prod_{i<j}(\alpha_i-\alpha_j)^2$. Where $\alpha_i$'s are roots of $f$.
0
votes
0answers
28 views

Proof number fields

In the proof Theorem 5.2 page 22 Janusz Number fields, that says Therem 5.2: The finite dimensional field extension $L$ of $K$ is separable if and only if the bilinear form $(x,y)=T_{L/K}(xy)$ from ...
5
votes
1answer
55 views

Proof that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a PID

How would one prove that $\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$ is a principal ideal domain (PID)? It isn't a Euclidean domain according to the Wikipedia article on PIDs.
1
vote
1answer
24 views

Prove that if $f\in R[X]$ , then $\displaystyle\prod _{\sigma \in G}f^{\sigma}\in \mathbb{Z}[X].$

Let $K$ be an algebraic number field and $R$ be the ring of algebraic integers of $K.$ Denote by $h^{\sigma}$ the polynomial obtained from $h\in K[X]$ after applying to its coefficients the ...
10
votes
3answers
171 views

likely open number theory problem: finite sum of $\zeta(2)$ equal to a square of rationals

Which $n$ can let $S=1+\frac14+\frac19+\cdots+\frac1{n^2}$ be a square of a rational number? Obviously, $1$ and $3$ work, but how to prove they are the only ones? I think this problem is really hard. ...
1
vote
0answers
61 views

$R_S (=K \cap A_{K,S})$ is a Dedekind domain

Let $K$ be a global field and let $S$ be a finite, nonempty set of places of $K$ containing the infinite ones. Show that $R_S (=K \cap A_{K,S})$, the ring of $ S-$ integers of $K$, is a Dedekind ...
2
votes
1answer
53 views

Periodicity over the prime indices of a multiplicative sequence implies periodicity?

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm ...