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Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

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Field $L$ over $K=\mathbb{Q}(\sqrt{-5})$ for which every $\mathfrak{p}\subset\mathcal{O}_K$ is unramified

If $K:=\mathbb{Q}(\sqrt{-5})$, find a non-trivial extension $L$ such that no prime $\mathfrak{p}\subset\mathcal{O}_K$ ramifies in $L$. I thought about using fact that every quadratic number field is ...
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Multiplication by an element inducing the identity on a quotient

I am studying the chapter on the associated Grössencharacter of a CM ellipic curve in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (II.9.) and have a question concerning a specific ...
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1answer
22 views

How to find the prime ideals of $\mathbb{Q}(\sqrt{2})$ in order to find ideals of norm $20$ (not prime).

Very new to algebraic number theory, so I was just wondering if someone could clarify how to do this? Essentially I would like to proceed by finding elements of norm 20. So $20=2^2\times5$ so I ...
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1answer
29 views

number fields with discriminant less than 8

Given the discriminant $|d_k| \le 8$. Determine all possible number fields of degree $n$. Using Minkowski I found the following bound on $n$ : $$\sqrt{|d_k|} \ge (\frac{n^n}{n!})(\frac{\pi}{4})^{n/2}...
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1answer
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Subextension of tamely ramified extension is tamely ramified

I know that if we have $L/K$ and $K'/K$ two extensions inside the algebraic closure $\bar K/K$, and $L'=LK'$. Then we have: $$L/K \:\text{tamely ramified} \Rightarrow L'/K' \:\text{tamely ramified}.$$...
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1answer
48 views

Decomposition and inertia fields in the factorization of $3$ in $\mathbb{Q}(\zeta_{24})$

I've seen the following exercise from an old problem sheet: For $\zeta:=\zeta_{24}$ a primitive $24$-th root of unity and $\mathcal{O}:=\mathbb{Z}[\zeta]$, determine the prime decomposition of $3$. ...
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28 views

Definition of the Norm Residue Symbol

Let $K$ be a field and $v$ its valuation. Let $K_v$ denotes the completion of $K$ with respect to the valuation $v$. If $L/K$ is a finite Galois extension, then all the $L_w$, for $w$ extending $v$, ...
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27 views

Why is that minimum positive element in the ideals of linear combinations is the GCD of it's factors?

Why is that the smallest positive element in the ideals of the form $a_1\mathbb Z + a_2\mathbb Z+...$ is the greatest common divisor of the coefficients $a_1, a_2...$? I have seen a proof of that ...
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The volume of $SL_2(\mathbf{Z}_p)$ is not $1$: question from Yu. Manin's paper “Reflection on Arithmetical Physics”

Yuri I. Manin in his paper "Reflection on Arithmetical Physics" gives an adelic proof of the celebrated Euler's formula $$\pi^2/6=\prod_p (1-p^{-2})^{-1} .$$He first consider the "adelic circle" $$A_{\...
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Factorising the ideal $(14)$

I wish to find the prime factors of the ideal $(14)$ in $\mathbb{Q}(\sqrt{-10})$. My working so far has been by noticing that $$14=(2+\sqrt{-10})(2-\sqrt{-10})=2\times7$$ So we have the candidates $...
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ternary quadratic forms over polynomial rings

The following theorem by Legendre is well-known: The integral ternary quadratic form $f(x,y,z) = ax^2 + by^2 +cz^2 \in {\bf Z}[x]$ represents $0$ non-trivially if and only if $a$, $b$, and $c$ do not ...
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35 views

Field of definition of isogeny of abelian varieties

Let $A/\mathbb{Q}$ be an abelian variety which is not simple over $\overline{\mathbb{Q}}$. Let $\phi$ be an isogeny (defined over some number field $K$) from $A$ to its geometrically simple components....
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1answer
24 views

Quadratic reciprocity and decomposition of primes in cyclotomic fields

In Neukirch's Algebraic Number Theory, there is a proof of the quadratic reciprocity which makes use of proposition $10.5$: $$p\text{ is totally split in }\mathbb{Q}(\sqrt{\ell^*})\Leftrightarrow p\...
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1answer
32 views

Other formulation for the discriminant

In the book of Frazer Jarvis on page 54 there is given a proposition about the discriminant . It is called $\textbf{Proposition 3.31 }$ Suppose that $K=\mathbb{Q(\gamma)}$ , and that the minimal ...
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224 views

New primality test for $2m^n+1$ (where $m$ is prime)? [on hold]

If $N=2.m^n+1$ (where $m$ is prime) you can prove if $N$ is prime or not by these two steps: Step (1) if $a^{2.m^{n-1}}=L \mod(N)$ (which is $L\neq1$ ) Step (2) $L^{m}=1 \mod(N)$ So N is prime. ...
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50 views

Fraction field and ring of integers

Let $K$ be a number field and let $O$ be a subring of the ring of integers $O_K$ of $K$. Show that $O$ contains a $\mathbb{Q}$-basis of $K$ if and only if the field of fractions of $O$ is $K$. I ...
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Showing that $3$ splits completely in $\mathbb{Q}(\sqrt{7}, \sqrt{19})$.

Let $K = \mathbb{Q}(\sqrt{7}, \sqrt{19})$, $L = \mathbb{Q}(\sqrt{7})$, and $M = \mathbb{Q}(\sqrt{19})$. Then we have that $3\mathcal{O}_L = \langle\sqrt{7} + 2 \rangle\langle2-\sqrt{7} \rangle$ and $...
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68 views

Solutions of $x^2+3y^2=p$ for $p$ prime [duplicate]

$x^2+3y^2=p \; \;$ for $p$ prime greater than $3$ has a solution if and only if $p\equiv 1\pmod 3$ I am supposed to use the fact that the class number of $\mathbb Q(\sqrt-3)$ is 1. I already got the ...
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Is $\mathbb{Z}[\sqrt{-5},1/10]$ a PID

Let $R = \mathbb{Z}[\sqrt{-5},1/2,1/5]$. Inverting the ramified primes $2,5$ simplifies the proof that every maximal ideal is inversible ie. the unique factorization in maximal ideals. In $O_K=\...
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37 views

Understanding surjectivity of $G_{\mathfrak{P}}\to G(\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p}))$

I'm trying to understand Theorem I.9.4 from Neukirch's Algebraic Number Theory (page 56). First he proves that $\kappa(\mathfrak{P})\mid \kappa(\mathfrak{p})$ is a normal extension, which is fine ...
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31 views

Hasse-Herbrand Function

Assume K is a finite extension of $\mathbb{Q}$$_p$. Let W = K (d$^{\frac{1}{p^n}}$), d $\in$ K$^{\times}$. And $\phi_{W/K}$ is the Hasse-Herbrand function of W/K. $\forall$ 1 $\leq$ j $\leq$ n, c$_j$ =...
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1answer
33 views

Galois group of residue field over $\mathbb Q$

Assume $L$ is a number field and is Galois over $\mathbb Q$,$\mathcal O_L$ is algebraic integer in $L$,$k_L=\mathcal O_L/p\mathcal O_L$ is the residue field,for a prime ideal $(p) \subset \mathbb Z$ ...
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1answer
55 views

Does this Galois group determine the class ideal class group?

Let $K$ be an algebraic number field, and consider the Galois group: $G = Gal(\bar{\mathbb{Q}}, K)$. Is knowing the Galois group $G$ alone, without other information on $K$, enough to determine the ...
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19 views

Normal basis of finite extension of a complete DVR

Let $R$ be a complete discrete valuation ring, and $S$ be a finite extension such that the associated residual field extension is separable. Then, why is it possible to choose a normal basis in powers?...
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78 views

A proof needed in minimal polynomial.

How do you prove this statement? If a, b are different prime numbers, the minimal polynomial of $\sqrt[n]{b}$ over the extension field $\mathbb{Q} (\sqrt[m]{a})$ of the rational number field $\mathbb{...
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1answer
39 views

A prime ideal $\mathbf{p}$ “lying above” a prime p

I am reading a book and we are considering an algebraic number field K, and its ring of algebraic integers O. We know that O contains the (usual) integers and in particular it contains the prime ...
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Precise factorization of Dedekind zeta function of cyclotomic field

Let $K=\Bbb Q(e^{2\pi i/m})$ be a cyclotomic field. I have frequently seen the assertion that its Dedekind zeta function $\zeta_K(s)$ factors into Dirichlet $L$-functions: $$ \zeta_K(s) = \prod_{\chi\...
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Morphism is étale

I am currently trying to understand a step in a proof concerning an étale morphism. Unfortunately I lack a deep understanding of this concept. I hope someone can elucidate that. Let $\phi: \mathbb{A}...
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1answer
36 views

Mordell's Equation Solutions

I need to solve the equation $y^3 = x^2+5$, factoring we get $y^3 = (x+\sqrt{-5})(x-\sqrt{-5})$. Now considering the two ideals $(x\pm \sqrt{-5})$, I should show that they are coprime. Now I've been ...
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Why is a pseudolattice $L$ a projective $\mathrm{End}(L)$-module of rank one?

If $L=L_\theta$ is a pseudolattice, that is, $L=\mathbb{Z}+\mathbb{Z}\theta$ where $\theta$ is an irrational real number, I know that $\mathrm{End}(L):=\{a\in\mathbb{R}\mid aL\subset L\}$ is an order ...
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1answer
32 views

find the generator of an ideal of an order

I'm reading Cohen's A Course in Computational Algebraic Number Theory, and I'm confused by the complexity of the algorithms for determining if an ideal of an order is principal, and finding its ...
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29 views

Invertible elements in a discrete valuation ring

I was starting to read 'Local Fields : Jean-Pierre Serre', and in the first section, it says that "the invertible elements of $A$ (a discrete valuation ring) are those elements that do not belong to $...
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Why Galois group of $K[ζ_{p^r}]/K[ζ_p]$ is cyclic?

Why Galois group of $K[ζ_{p^r}]/K[ζ_p]$,p is a odd prime,is cyclic? Why Galois group of $K[ζ_{2^r}]/K[ζ_4]$ is cyclic? ζ denotes a primitive root of unity.K is a number field. These facts are used in ...
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1answer
85 views

A special inverse Galois problem

Suppose that $G$ is a transitive permutation group and suppose we know a construction of an isomorphism from the Galois group of a Galois number field to $G$. Does this information make it easier to ...
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1answer
50 views

Marcus Number Fields Chapter 4 Exercise 8

Let $r,e,f$ be given positive numbers. I should try to demonstrate that there are always $p,q$ prime such that $p$ splits exactly in $r$ different prime in the $q$th cyclotomic field. I know that ...
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3- class group of Pure cubic field

I would like to study more about 3-class group of pure cubic field. Can any one suggest me some good reference book/article/research papers.
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2answers
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Inverse of the principal ideal $(\sqrt{-15})$

We work in the subring $R= \mathbb{Z} + \mathbb{Z}(\frac{1 + \sqrt{-15}}{2})$ of $\mathbb{Q} + \mathbb{Q}(\frac{1 + \sqrt{-15}}{2})$. The exercise asks us the give the number of principal ideals of ...
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1answer
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Is it true that $f(e) \ne g(\pi) \,\,\forall f,g$ are non zero algebraic functions

Just a random question that popped in my head. This is of course non true if we consider just irrationals. Moreover, (assuming above holds) does there exists some sort of equivalence class for ...
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27 views

Images on tangent spaces generate Lie(A)

I am reading this paper: https://gdz.sub.uni-goettingen.de/id/PPN356556735_0088?tify=, but have quite little knowledge about algebraic groups and their Lie-algebras. I have a question about the ...
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absolute value, sup(v(x+1)) bounded if $\mathbb{N}\cdot K$ is bounded

Let $K$ be a field and $v: K \mapsto \mathbb{R}_+$ a map such that $v(x)=0 \Leftrightarrow x=0$ and $v(x \cdot y)=v(x) \cdot v(y)$. Assume that $v$ is bounded on $\mathbb{N}\cdot 1 \subset K$. Is it ...
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Statement of Artin Reciprocity by Silverman

I have a seemingly easy question concerning the Artin Reciprocity as stated in the book Advanced Topics in the Arithmetic of Elliptic Curves by Silverman. He states it as follows. Let $L/K$ be a ...
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2answers
84 views

Extensions of Chicken Nugget Theorem

Given that $a,b,c\ $ are pairwise relatively prime, what is the largest number not expressible as a linear combination of $a,b,c\ $? What if $a$ and $b$ have common factor $m$? What if $a$ and $b$ ...
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1answer
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If $[K:\Bbb{Q}]=2$ then $K=\Bbb{Q}(\sqrt{d})$.

I am stuck on one question and sincerely have no idea how to proceed. Let $K$ be a field containing $\Bbb{Q}$ such that $[K : \Bbb{Q} ] = 2$. Prove that there exists a square free integer $d$ such ...
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1answer
38 views

$D = \mathbb{Z} + \mathbb{Z} \sqrt{d}$ where $d \equiv 1 \bmod 4$, show that the principal ideal $2D$ is ramified.

$D = \mathbb{Z} + \mathbb{Z} \sqrt{d}$ where $d \equiv 1 \bmod 4$. Show that there is a unique prime ideal of $D$ that contains the principal ideal $2D$. I would like help showing $2D$ is ramified, ...
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1answer
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How many solutions of the equation $ax^2 +by^2 = 1$ are there with $(x, y) ∈ \mathbb{F}_{p} ×\mathbb{F}_{p}$ [duplicate]

How many solutions of the equation $ax^2 +by^2 = 1$ are there with $(x, y) ∈ \mathbb{F}_{p} ×\mathbb{F}_{p}$ where $a, b$ are integers whose product is not divisible by $p$? This was a recommended ...
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0answers
47 views

Explicit calculation of residue fields in quadratic extensions

Let's say we have a prime number $p=5,$ and we are interested in how the residue field $\mathcal{O}_k/P$ looks like, where $P$ lies over 5 in quadratic extension $k.$ I suppose this has to depend on ...
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1answer
83 views

Unramified nonabelian extension of number field with class number 1

Let $K$ be a number field. Global class field theory tell's us that if the class number of $K$ is 1, then there's no unramified abelian extension (including archimedean places) of $K$. But I'm curious ...
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25 views

An equation defined by norm

Let $f$ be an Eisenstein polynomial of degree $n$ and the prime $p$. $\alpha$ is a root of $f$. Let $\mathbb{Q}(\alpha)=K$, Prove that for any $\gamma\in O_K$, there exist $a\in \mathbb{Z}$, such that ...
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1answer
42 views

The set of primes that split $x^2-5$ is $\{p:p \equiv \pm1 \mod 5\}$.

My professor worked this out in class and I am lost on how he did it. The definition I am using for split is, for a polynomial $f$ with degree d, with integer coefficients, a prime is $f$-split if $f\...
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1answer
34 views

Proper Ideals with Norm Relatively Prime to Conductor

Let $K$ be an imaginary quadratic number field, and $\mathcal{O}_K$ the ring of integers. Let $\mathcal{O}$ be an order. Call the $\textit{conductor}$ $f = [\mathcal{O}_K:\mathcal{O}]$. Given some $\...