Questions related to the algebraic structure of algebraic integers

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Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. ...
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1answer
40 views

zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : ...
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42 views

A question about square roots of quadratic residues.

Suppose $\mathbb{Z}_p^*$ ($p$ is a prime) is a cyclic group with generator $g$. We consider a subgroup $\mathbb{G}$ of $\mathbb{Z}_p^*$ with generator $h$ and order $q$, where $h = g^4~mod~p$ and ...
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Properties of the norm in a Euclidean Domain

I am aware of the fact that the Euclidean Norm does not need to be unique in a given domain, however my question is essentially: can we ensure that the properties of the norm remain the same? More ...
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35 views

Non-existence of a particular type of tower of number fields

I have number fields $\mathbb{Q}\subset K\subset H$ where $K\subset H$ is Galois. I want to show that is is impossible for a rational prime $p\in\mathbb{Z}$ to remain first inert in $K$ but then for ...
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What are some practical attempts to disprove Riemann Hypothesis?

Most people believe Riemann Hypothesis is true. Since RH has not been proved yet, so it is not completely insane to disprove RH. Among the ways to disprove RH, straightforward ways, such as: try to ...
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1answer
31 views

Determining when ring of integers is $\mathbb{Z}[\theta]$

Something which is not difficult to prove is that if $K$ is a number field generated by an integer $\theta$, then the ring of integers $\mathfrak{O}_K$ is generated over $\mathbb{Z}$ by $\theta$ and ...
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1answer
15 views

Confusion about definition of primitive polynomials

I am working through Neukirch's Algebraic Number Theory and am confused about his definition of primitive polynomials on page 129. He defines $f(x)=a_0+a_1x+\dots +a_nx^n$ on $\mathcal{O}$ with ...
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2answers
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Can we describe nicely all the rational numbers of the form $x^{2}-xy+y^{2}$?

Let $\zeta$ be a primitive cubic root of unity, i.e., $\zeta$ is a complex number such that $\zeta^{3}=1$ and $\zeta\neq1$. Let us consider the Galois extension $\mathbb{Q}(\zeta)/\mathbb{Q}$ (the ...
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2answers
19 views

ramification of prime in Normal closure

Let $K$ be an algebraic number field and let $p$ be a prime in $\mathbb{Q}$ such that $p$ ramifies in $L$, the Galois closure of $K$. How can I show that $p$ ramifies in $K$ itself?
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55 views

Unramified algebraic extensions of local fields

This is a basic question from Neukirch's Algebraic Number Theory, Prop. 7.2: Fix a non-Archimedean local field $K$. Let $L/K$ and $K'/K$ be two extensions inside an algebraic closure $\bar{K}/K$ and ...
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34 views

what is the volume of $cos\pi \theta$? [duplicate]

I wana prove that if a=cos$\pi\theta$ is rational number(and also assume $\theta$ is rational,too) it can be just {-1/2,1/2,1,-1,0}.Before this, I proved that a is an algebraic number and I know that ...
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1answer
114 views

Factors of the numbers of the form $a^2+nb^2$

Let $N=a^2+nb^2$ with $\gcd(a,b) =1$ and $n \in \mathbb{Z^+}$. If $N=xy$ where $x$ and $y$ are relatively prime numbers, in what condition can $x$ and $y$ be also written in the same form as $N$ ...
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1answer
64 views

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
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1answer
94 views

$\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field.

The problem I'm facing is that of the tittle: Problem. Prove that $\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field. Since $23\not\equiv 1\pmod{4}$, it must be shown that ...
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1answer
32 views

What are the units in $\mathbb{Z}[\root 3 \of 2]$?

I asked Wolfram Alpha to tell me the fundamental unit of $\mathbb{Z}[\root 3 \of 2]$, it replied $1 - \root 3 \of 2$. Then I tried asking it for $(1 - \root 3 \of 2)^n$ for $-5 \leq n \leq 5$. If I ...
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1answer
74 views

Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
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34 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 ...
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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: $\mathfrak{p}\mathcal{O}_L=\mathfrak{P}_1^{e_1}\dots\mathfrak{P}_g^{e_g}$. ...
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39 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 ...
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1answer
49 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 ...
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18 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 ...
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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 ...
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43 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 $L$ ...
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1answer
34 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?
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2answers
38 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 ...
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0answers
12 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 ...
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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 $S = \sum_{x=0}^{p-1} \psi(x^2 - a)$, ...
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Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
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1answer
97 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 ...
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2answers
72 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 $\mathcal{O}_F$?
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Are the sets of valuations uniquely determined

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 $K$?
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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 ...
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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 ...
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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 ...
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2answers
59 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 ...
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75 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 ...
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2answers
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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, ...
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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} ...
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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. ...
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1answer
36 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 ...
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1answer
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$a(x)$, $b(x) \in \mathbb{C}(x)$ 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 ...
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1answer
37 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 $A$, ...
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17 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 ...
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1answer
53 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). ...
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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 ...
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3answers
50 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 ...
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1answer
108 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 ...
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1answer
39 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 ...
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1answer
32 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$ ...