Questions related to the algebraic structure of algebraic integers

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3
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1answer
35 views

Show that $\xi^3\equiv \pm 1 \pmod{\lambda^4}$ in $\Bbb Z [\omega]$

We have $\lambda=1-\omega$ where $\omega=e^{i 2\pi/3}$ and $\xi$ an Eisenstein integer. Given that $\xi \equiv \pm 1 \pmod{\lambda}$, how can I prove that $$\xi^3\equiv \pm 1 \pmod{\lambda^4}$$ I ...
1
vote
0answers
51 views

An problem of ideal splitting in number field extension

If $L/K$ is Galois extension of number field, $\mathfrak{p}$ is an prime integral ideal of $K$. One would asserts that: $\mathfrak{p}\mathcal{O}_L=\mathfrak{P}_1^{e_1}\dots\mathfrak{P}_g^{e_g}$. ...
2
votes
0answers
44 views

Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
1
vote
1answer
58 views

Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is Q[$\sqrt{2}+\sqrt{3}$]=Q[...
0
votes
0answers
28 views

Ratio of maximum and minimum value

I've tried this so far. Let no. of $-1s$ are $a$, no. of $1s$ are $b$ and no. of $2s$ are $c$. Now $-a+b+2c=19, a+b+4c=99$ On adding and subtracting the equations $a= 40-c$ and $b=59-3c$ Now $x_1^3+...
1
vote
0answers
32 views

Computing ideal class group by other means than the Minkowski bound?

When calculating the ideal class group of a number field, it is common to start with the Minkowski bound, followed by decomposing finitely many prime ideals of norm less than that bound, and finding ...
7
votes
1answer
54 views

Unramified primes of splitting field

I would like to show the following: Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ ...
0
votes
1answer
35 views

A diophantine equation of degree 3

Find the integer solutions of $y^2+6=x^3$. I guess it does not have integer solutions but I cannot prove it. By $\pmod 8$, I can know that $y$ is odd and $x\equiv7 \pmod 8$. Then what else can I do?
2
votes
2answers
41 views

Question about S.Lang's proof of Kummer's Lemma

I have a question about the proof of Kummer's Lemma in Serge Lang's Cyclotomic fields (i.e. Theorem 6.1). Let $K = \mathbf{Q}(\xi_p)$ the $p$-th cyclotomic field extension of $\mathbf{Q}$. Let $u$ be ...
0
votes
0answers
14 views

Automorphisms of local field

(1)Suppose that $K$ is a local field but not $\mathbb C$. Then show that any automorphism of $K$ is continuous. (We can assume that $K$ is $\mathbb R$ with classical absolute value or $K$ is a finite ...
2
votes
0answers
27 views

Dirichlet character of order $4$ and the splitting of $p$ in $\mathbb{Z}(\sqrt{-1})$

For $p \equiv 1 \pmod{4}$, let $\psi$ be one of the two Dirichlet characters of order $4$ in $(\mathbb{Z} / p \mathbb{Z}) ^\times$. Consider the character sum $S = \sum_{x=0}^{p-1} \psi(x^2 - a)$, ...
1
vote
1answer
35 views

Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
4
votes
1answer
109 views

Is $3$ prime in the ring of integers of the field $\mathbb{Q}(\sqrt{2\sqrt{2}-1})$?

I am trying to determine if the number $3$ stays prime in the ring of integers of the quartic field $K=\mathbb{Q}(\sqrt{2\sqrt{2}-1})$, or rather adjoin a real root of $X^4+2X^2-7$. I do know that $...
4
votes
2answers
77 views

Algebraic number fields in which all rational primes are inert

Is there an algebraic number field $F\supsetneq\mathbb{Q}$ such that all rational primes are inert in $\mathcal{O}_F$?
0
votes
1answer
20 views

Are the sets of valuations uniquely determined [closed]

Let $\mathcal{V_K}$ be the set of valuations of a number field $K$. Can it be that $\mathcal{V_L}=\mathcal{V_K}$, for the set of valuations of another number field $L$ non-isomorphic to $K$?
0
votes
1answer
26 views

Completions of number fields at the same prime

This is probably obvious, but I don't quite see it. Archimedean completions of different number fields are always isomorphic to the same $\mathbb{R}$ or $\mathbb{C}$. Is the same true in the non-...
1
vote
1answer
25 views

Index of norm group of a global field

For a global field $K$ (characteristic $p$ or $0$), is there anything meaningful which could be said about the value $[K^\times:N_{L/K}L^\times]$ where $L$ is a finite extension (possibly of prime ...
2
votes
0answers
75 views

How calculate Galois Group of $\mathbb{Q}_7(\zeta_3,\sqrt{3})/\mathbb{Q}_7$

From local field theory I know that if $L/K$ is a Galois extension of number fields, $\mathfrak{P}$ is a prime ideal of $L$ living above a prime $\mathfrak{p}$ of $K$, then the extension $$L_{\...
4
votes
2answers
68 views

Convert from Nested Square Roots to Sum of Square Roots

I am looking for a way to easily discover how to go from a nested root to a sum of roots. for example, $$\sqrt{10-2\sqrt{21}}=\sqrt{3}-\sqrt{7}$$ I know that if i set $\alpha=\sqrt{10-2\sqrt{21}}$,...
2
votes
0answers
79 views

Any natural number n can be expressed as $n = 2^a \cdot b$ where $b$ is odd. Function such that $f(n) = a$

Given that any natural number $n$, can be expressed $n = 2^a \cdot b$ where $b$ is odd. Is there a function that does not include modulo or floor functions that satisfies $f(n) = a$? Thus far I have ...
1
vote
2answers
64 views

Norm and trace of $\sqrt{15}$ over $K = \Bbb Q(\sqrt3, \sqrt5)$

I have been stuck on an algebraic number theory question, could you please show me how you would approach this: work out the norm and trace of $\sqrt{15}$ over the number field $K = \Bbb Q(\sqrt3, \...
3
votes
0answers
45 views

How else can I tell I can do this with $5$ but not $2$ or $3$ in $\textbf{Z}[\sqrt{30}]$?

In $\textbf{Z}[\sqrt{30}]$, the number $5$ splits, since, for example, $N(5 + \sqrt{30}) = -5$. But the ideal $\langle 5 \rangle$ is a ramifying ideal, since it is equal to $\langle 5, \sqrt{30} \...
6
votes
0answers
74 views

An equality in the proof of Proposition 3 of Section 2.7 of Pierre Samuel's Algebraic Theory of Numbers

I am reading Pierre Samuel's Algebraic Theory of Numbers. I get stuck at an equality within the proof of Proposition 3 of Section 2.7. The statement of the proposition is as follows: Proposition 3. ...
3
votes
1answer
42 views

Function field, finite extension, isomorphism implies isomorphism?

Let $A$ be a function field in $1$ variable over $\mathbb{C}$, and let $B$ be a finite extension of $A$ of degree $[B : A]$. If $B \cong \mathbb{C}(x)$ over $\mathbb{C}$, then does it necessarily ...
4
votes
1answer
50 views

$a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$ implies $a(x)$, $b(x)$ constant?

If $a(x)$, $b(x) \in \mathbb{C}(x)$ and $b(x)^2 = a(x)^3 + 1$, then does it necessarily follow that $a(x)$ and $b(x)$ are constant? Edit. To clarify, $\mathbb{C}(x)$ is the field of rational ...
4
votes
1answer
40 views

Criterion for the integral closure of an domain in a finite field extension being a finitely generated algebra

$A$ is an integral domain, $K=\operatorname{Frac}A$, $L/K$ finite field extension (not necessarily separable), $B$ is the integral closure of $A$ in $L$. Question: with some extra conditions on $A$, ...
0
votes
0answers
23 views

Proof for Neukirch mysterious relationship between frobenius elements in abstract CFT

In Neukirch's ANT chapter (4) on Abstract Class Field Theory, there is a claim which I can prove, but I can't prove the "In particular" part that follows. I'm stuck in this for almost a week. I've ...
2
votes
1answer
55 views

There are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field

My book's exercise is about proving that there are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field (with respect to the norm). ...
0
votes
0answers
23 views

Norm of ideal belongs to the ideal [duplicate]

Suppose that $D$ is any number ring (i.e. $D=\mathbb Q(\alpha), \alpha \in \mathbb C$). Let $I$ be any ideal of $D$. Show that $N(I)=|D/I|$ belongs to $I$. How to start? is there a specific fact will ...
3
votes
3answers
55 views

How to find open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$?

For purposes of illustrating Local Class Field Theory, let us play with the $3$-adic numbers. I'd like to find some open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$. I know about the ...
4
votes
1answer
114 views

A line in a proof regarding nth power residues

I would appreciate help understanding this highlighted line in a proof in Ireland & Rosen (p. 45). I don't know much group theory although I know the residue classes $\pmod m$ form a ...
0
votes
1answer
41 views

A subgroup $Z$ of $p$-adic integer such that $Z/pZ\neq C_p$.

This question may be weird in the sense that I am considering a subgroup of infinite index. Let $p$ be a prime (let's say $p\neq 2$ although I'm not sure whether this assumption is needed) and we ...
2
votes
1answer
39 views

Using Dedekind's prime ideal factorisation theorem

I've been going over past papers for algebraic number theory and came across this question which has given me some trouble: Given a number field $K =\mathbb{Q}(\sqrt{-d})$ where $ d\equiv 1 \mod 4$ ...
0
votes
1answer
19 views

Group $U$ of $p$-adic units is inverse limit of $U/U_{n}$

In Serre's famous Course in Arithmetic, there is a somewhat unexplained claim: Let $U=\mathbb{Z}_{p}^{\times}$ be the group of $p$-adic units. For every $n\geq 1$, put $U_{n}=1+p^{n}\mathbb{Z}_{p}$...
2
votes
0answers
29 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
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0answers
23 views

Problem when computing ideal class group

When computing the ideal class group of a quadratic extension $\mathbb{Q}[\alpha]$ after we have decomposed all rational primes smaller than the Minkowski bound into generating prime ideals $\mathfrak{...
2
votes
0answers
29 views

Neukirch ANT - Reciprocity map doesn't depend on primes

This is a quick question. Unfortunately, Neukirch does CFT in such a unique way that it is not very easy to explain all the notation and concepts he defined here, so I expect someone who read the book ...
1
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0answers
25 views

Computing $n^{\text{th}}$ root of a positive integer to arbitrary precision using integer arithmetic

There are various questions on this forum that appear similar, but my question pertains to writing code that can compute the $n^{\text{th}}$ root of a number $a$ correct to $p$ decimal places, where ...
1
vote
1answer
41 views

Isomorphism of completions of number fields

Let $K$ and $L$ be number fields, $v$ a place of $K$ (either archimedean or non-archimedean) and $\theta:K\simeq L$ a ring isomorphism. I am trying to show that $\theta$ induces an isomorphism $K_v\...
1
vote
2answers
39 views

Completeness proofs for the solutions of Diophantine Equations

In general, what are the strategies for showing the completeness of a solution set for Diophantine Equations? For example, take the $\textit{Pell-type}$ equation $x^2 - dy^2 = a$. Say, you have a set ...
3
votes
2answers
81 views

Significance of the Riemann hypothesis to algebraic number theory?

Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? Moreover,...
2
votes
1answer
42 views

Ramified primes in radical extension of number fields

Let $ K $ be a number field, $ n\ge2$ be a positive integer and $a \in K^*$. How does one show in the simplest possible way that a prime ideal $\mathfrak {p}$ of $ K $ not dividing $ n$ is ...
2
votes
0answers
25 views

Adelic definition of “canonical divisor”

For a function field over a curve $F/K$, some book define the canonical divisor as the divisor of a map $\omega:\mathscr{A}_{F}\rightarrow K$ (where $\mathscr{A}_{F}$ is the pre-adele, ie. adele but ...
0
votes
0answers
23 views

Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
1
vote
1answer
32 views

formal derivative algebraic [closed]

Let $q$ be a prime power, $\mathbb{F}_q$ the field with $q$ elements and $f \in \mathbb{C}_\infty$ be of the form $f = \prod_{i=1}^\infty f_i$ with $f_i \in \mathbb{F}_q(X)$ (here $\mathbb{C}_\infty$ ...
1
vote
1answer
18 views

Splitting of a prime in the compositum of two fields [duplicate]

Let $L$ and $M$ be two finite extensions of $\mathbb{Q}$ and let $LM$ denote their compositum. Suppose that $p$ is a rational prime that splits completely in $L$ and $M$. How can I show that $p$ ...
0
votes
1answer
68 views

In $\mathbb{Z}[\omega]$, if $a^3+b^3+c^3=0$ then $1-\omega$ divides at least one of $a,b,c$

This is problem 3.26 (self-study) in "Ireland and Rosen" If $a,b,c \in \mathbb{Z}[\omega]$ and none are equal to zero, and $a^3 + b^3 +c^3 = 0$ , show at least one of $a,b,c$ is divisible by $...
0
votes
1answer
41 views

A question about the definition of $p$-adic pseudo-measure.

Let $\mathfrak B$ be a profinite abelian group and let $\Lambda(\mathfrak B)$ be defined as the inverse limit $\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$ where the inverse limit is taken with ...
2
votes
1answer
64 views

Proving an equivalence relation, $a^3\equiv 1 \pmod 9$, in $\mathbb{Z}[\omega]$

I would appreciate help with two steps in solving this problem (self-study) from Ireland & Rosen (3.25) The problem states: Let $\lambda= 1-\omega \in \mathbb{Z}[\omega]$. And $a\equiv 1\pmod \...
3
votes
1answer
33 views

Computing prime factorization of ideals?

I want to compute the prime factorizations of the ideals $\langle 4\sqrt{-14}\rangle$, $\langle 6\sqrt{-6} \rangle$ and $\langle 4\sqrt{-5} \rangle$ in the ring of algebraic integers of $\mathbb{Q}(\...