Questions tagged [algebraic-number-theory]

Questions related to the algebraic structure of algebraic integers

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Understanding discriminant of composite field via representation theory

Let $L, K$ be two number fields. If $L$ and $K$ are linearly disjoint over $\mathbb{Q}$, then we know (by [Neukirch, ch.I (2.11), Proposition]) $$d_{LK}=d_L^{[LK:L]}d_K^{[LK/K]}$$ where $d_K$ stands ...
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Find the number of order triples (x, y, z) of integers?

Question: Find the number of order triples x, y, z of integers such $that$x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=24$My try:$(x^2-y^2-z^2)^2=x^4+y^4+z^4-2x^2y^2+2y^2z^2-2x^2z^2$So,$x^4+y^4+z^4-2x^2y^...
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What do we know about integer solution(s) to $x^p+ay^p=1$, where $p$ is a prime number and $a$ is a fixed integer?

We know how to write the solution to the equation $x^2+ay^2=1$. Due to a Theorem by Nagell the solution to $x^3+ay^3=1$ has at most two solutions. What do we know about $x^p+a y^p=1$. Does this have ...
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When and how does a rational prime (not ideal!) reduce in a given number field?

I apologize if my abstract algebra is so shaky that I might have glossed over an answer to this problem. Yesterday I suddenly got the motivation to do some number theory again - as a challenge, I ...
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Does minimal polynomial being separable mod $\mathfrak p$ imply that $p$ does not divide $\left|\mathcal O_L / \mathcal O_K[\alpha]\right|$?

Let $L/K$ be a Galois extension of number fields, where $L=K(\alpha)$ for some algebraic integer $\alpha\in\mathcal O_L$. Let $f(x)$ be the minimal polynomial of $\alpha$ over $K$. Let $\mathfrak p$ ...
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Irreducibility of $x^8+6x^4+1$

Is there a simple way to show that the polynomial $P(x) = x^8+6x^4+1$ is irreducible over $\mathbb Q$ by hand? The CAS software Sage tells me that it indeed is. It appears as polynomial 8.0.4194304.1 ...
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1 vote
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Question about Neukirch ANT Proposition $4.2$

A subgroup $\Gamma\subset V$ is a lattice if and only if it is discrete. The proof is on page 25 there are few non trivial lines for me As the $\mu_i= \gamma_i- \gamma_{0i}\in \Gamma$ Why $\mu_i$ ...
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Can you deduce the size of a residue field of a number field with a $P$-adic norm based on its degree and $p$ that $P$ lies over?

Let $K$ be an algebraic number field with a $P$-adic norm and $\mathcal{O}_K$ it's ring of integers. We know that by definition $K$ is an extension of $\mathbb{Q}$ of degree $n$. We also know that $P$ ...
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Doubt on Neukirch ANT $(3.1)$ theorem .

On the page 17 of neukirch ANT i have a doubt in the proof The ring $\mathcal{O}_K$ is noetherian, integrally closed, and every prime ideal $\mathcal{p}\neq0$ is a maximal ideal In the proof while ...
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Doubts in Silverman's AEC Chapter 3 Prop 1.5

On Page $48$ of Silverman's Arithmetic of Elliptic Curves, he proves the theorem that the invariant differential associated to a Weierstrass equation for an elliptic curve is holomorphic and ...
1 vote
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Error in proof of Childress Class Field Theory, Theorem 7.2

Let $E = \mathbb{Q}(\zeta_p)$, and $E^+ = \mathbb{Q}(\zeta_p + \zeta_p^{-1})$. Let $C_K$ denote the class group of a number field $K$. Theorem 7.2 of Childress states that the map $C_{E^+} \to C_E$ ...
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Is $K^\times/F^\times N_{L/K}(L^\times)$ infinite when $L/F$ is a biquadratic extension of a number field $F$?

Let $F$ be a number field, let $L=F(\sqrt{\alpha_1},\sqrt{\alpha_2})$ be a biquadratic extension, and let $K=F(\sqrt{\alpha_1\alpha_2})$. Question. Is is true that $K^\times/F^\times N_{L/K}(L^\times)$...
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Existence of prime ideals with given splitting in biquadratic extensions

Let $F$ be a number field, let $L=F(\sqrt{\alpha_1},\sqrt{\alpha_2})$ be a biquadratic extension of $F$ (so $\alpha_1,\alpha_2$ and $\alpha_1\alpha_2$ are not squares in $F$) Does there exists a prime ...
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1 vote
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Understanding the proof of the finiteness of the class number

The proof is in Chapter 12 of Rosen and Ireland's A Classical Introduction To Modern Number Theory. Proposition 12.2.3 states that $D/A$ is finite, where $D$ is the ring of all algebraic integers in a ...
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What is going on with the discriminant of $f^{\circ n}(X)-X$?

Recently, I found this amazing answer of @Mercio where they point out the following about the discriminant of the polynomial $P(X)=\frac{f^{\circ 3}(X)-X}{f(X)-X}$ where $f(X)=X^2+c$. We can compute ...
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Understanding proof of existence theorem of global class field theory more step by step ( Neukirch, Algebraic number theory VI - (6.1) theorem )

I am reading the Neukirch, Algebraic number theory, p.395, proof of (VI)- (6.1) theorem and stuck at some statements : First, we equip the idele class group $C_K$ with its natural topology ( C.f. his ...
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Group generated by Image of Uniformizer under local Artin Map

Let $L_w/K_v$ be a finite abelian extension of complete non-archim valued local fields and $\Phi: K_v^{\ast} \to Gal(L_w/K_v)$ canon induced from local Artin map (composed with qoution on codomain ...
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Why isotropic pairing $〈, 〉$ induces　$B\cong B^*\stackrel{f^*}{\to}A^*$ and $B\to \text{Coker}f \hookrightarrow A^*$ coincides as a map

Let $A,B$ be abelian groups. $A^*$ be its dual, that is, $A^*=\text{Hom}(A,\Bbb{Q}/\Bbb{Z})$. Suppose there is a non-degenerate pairing $〈,〉: B\times B \to \Bbb{Q}/\Bbb{Z}$ and this pairing induces an ...
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Does $\alpha-\alpha^r \in K$ imply $\alpha \in K$?
Let $K$ be a finite extension of the $p$-adic number field $\mathbb{Q}_p$. Assume two algebraic numbers $\alpha-\alpha^r \in K$ and $p\beta-\beta^{p^r} \in K$, where $r \in \mathbb{N}$ and \$\alpha,~\...