Questions tagged [ramification]

Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors.

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A question about the inertia index of prime ideals in $\mathbb{C}(t)$

Consider the field extension $[F:\mathbb{C}(t)]=d$. Now consider $A$, the integral closure of $\mathbb{C}[t]$ in $F$. Why is it true that every prime ideal $\mathfrak{p}\subset \mathbb{C}[t]$ has ...
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Contradicting arguments regarding the ramification index of the splitting field of $X^3 - 135X - 270$ over $\mathbb{Q}_5$

Let $f = X^3 - 135X - 270 \in \mathbb{Q}_5[X]$ and $L$ the splitting field of $f$ over $\mathbb{Q}_5$. Let $e$ be the ramification index of $L/\mathbb{Q}_5$. If I am not mistaken, the degree of $L/\...
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Maximal tamely ramified abelian extension of $\mathbb{Q}_p$ is finite over the maximal unramified extension $\mathbb{Q}_p^{nr}$?

I came across an curious exercise in Neukirch's algebraic number theory book. Exercise 2 page 176 (Chapter II section 9) asks the following Prove that the maximal tamely ramified abelian extension V ...
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Tamely ramified extensions.

I have a question about a result on tamely ramified extensions in Neukirch's Algebraic Number Theory. Proposition 7.7 in chapter II section 7. The question I have is about the proof which starts by ...
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If $E$ has additive reduction at $v$ then $H^0(I_v,E[p^\infty])$ is finite

Let $E$ is an elliptic curve defined over a number field $F$, $p$ a prime of $\mathbb{Z}$ and $v$ a valuation of $F$ that does not lie over $p$. Call $F_v$ the completion of $F$ with respect to $v$ ...
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On a tamely ramified extension of $\mathbb{Q}_{p}$

I'm stuck with the following problem given in a book which I'm reading, it's about creating a tamely ramified extension of $\mathbb{Q}_{p}$. Let $p\in\mathbb{Z}$ be a prime number, and let $\mathbb{Q}...
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Ramifications indices invariant under algebraic closure + question about ramification and covering

We are considering functions fields of transcendance degree equal to $1$ (or smooth curves, it's the same), over a perfect field if necessary, or even a finite fields, let's say $K(X)$ and $K(Y)$ (...
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Completion with respect to conjugate valuations

Let $K$ be a number field and $L/K$ an algebraic Galois extension. If $v_{P}$ is the place of $K$ corresponding to the prime ideal $P$, then we know that there are valuations of $L$ extending $v_{P}$. ...
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Motivation for considering the upper numbering of ramification groups

Let $L/K$ be a finite Galois extension. We denote by $G_s$ its $s$-th ramification group. Define the Herbrand function $$\eta_{L/K}:[-1,\infty) \to [-1, \infty), \ \eta_{L/K}(s) = \int_0^s \frac{1}{(...
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Etale Ring Homomorphism

Let $R$ be a complete ( not sure if it's neccessary) DVR together with a ring $h: \mathbb{Z} \to R$ and $p \in \mathbb{P}$ a prime number. Let $f: R \to S$ a finite ring homomorphism. We assume two ...
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Existence of Prime Factor with Ramification Index 1

I have been studying some basic ramification theory, and, given the standard "$AKLB$ configuration", - where $A$ is a Dedekind Domain with field of fractions $K$, $L$ a finite separable extension of $...
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Serre's local field, Chapter 1 Proposition 10.

The proposition: Let $p$ be a non-zero prime ideal of $A$, the ring $B/pB$ is an $A/p$ algebra of degree $n=[L:K]$, isomorphic to the product $\prod_{P\mid p}B/P^{e_P}$. We have the formula $n=\...
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If $\rho$ is a mod $p$ irreducible representation of $G_{\mathbb{Q}_p}$, why is it tamely ramified?

Fix a prime $p$ and let $\rho:G_{\mathbb{Q}_p}\rightarrow GL_2(\overline {\mathbb{F}}_p)$ be an arbitrary continuous representation. I found the following statement in a paper on non-ordinary modular ...
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Question about inertia groups and unramified extensions

Let $K$ be a number field, and $v$ a finite place. If $\bar{K}$ is a separable closure of $K$, then in $G_K=\text{Gal}(\bar{K}/K)$ we can find the decomposition group of (a place over) $v$, which is ...
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Ramification in Maximal Totally Real Subfield

Exercise 12, Chapter 4 of Marcus' Number Fields asks the following: Let $p$ be a prime not dividing $m$. Determine how $p$ splits in $\mathbb{Q}(\zeta_m +\zeta_m^{-1})$. Certainly $p$ is unramified, ...
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Prime splitting in imaginary squarefree quadratic field

This appeared on my number theory final. I couldn't figure it out. The setup is: Let $K = \mathbb{Q}[\sqrt{-d}]$ be a quadratic number field, with $d = p_1...p_r$ a product of $r$ distinct odd ...
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Show that a prime is ramified in a cyclotomic field if and only if it divides the discriminant, without using relative discriminants

I want to show that a rational prime $p$ is ramified in a cyclotomic field $K = \mathbb{Q}(\zeta_q)$, where $q$ is an odd prime, if and only if it does not divide the discriminant $\Delta_K$. I have ...
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branching Galois covering space degree of the point

Let $\pi: R \to S$ be a branching Galois covering space of degree $d$. Show that for each branch point the branch order of the point is a divisor of $d$. Our definition: A $Galois$ covering is a ...
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Equivalent condition for an extension of local fields to be unramified

Let $L/K$ be a degree $n$ finite Galois extension of local fields with corresponding residue field extension $\ell/k$ having residue class degree $f$ and ramification index $e$. We have a canonical ...
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Ramification group as unique p-Sylow subgroup

Let $L/K$ be a finite Galois extension of valued fields with unique extension of the valuation from $K$ and $G_s$ be the $s$-th ramification group. One can show that $G_0/G_1 \hookrightarrow \lambda^{\...
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Reduction step for studying cyclic algebras over local fields

I am trying to study cyclic algebras $(\chi,a)$ over a local field $K$ (specifically of characteristic 0). Given a cyclic Galois extension $L/K$ with isomorphism $\chi\colon Gal(L/K)\rightarrow\mathbb{...
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Algebraic varieties : How to calculate explicitly the ramification index?

Again a question about algebraic varieties ! Actually, I followed to book of Silverman "The Arithmetic of elliptic curve", and I have several questions about ramification index. For $\phi : C_1 \to ...
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Unramified morphism of schemes is locally of finite presentation

Let $f:X\to Y$ be a morphism of schemes and $x\in X$. Is it true that $f$ unramified in $x\Rightarrow f$ is locally of finite presentation? If yes, I don't see how to prove it? Thank you for your ...
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Separability in the Definition of Ramification

Maybe this question will be clearer to me once I've read more about algebraic number theory but conversely maybe an answer to this question will help me at doing so. Let $\mathcal{O}$ be a Dedekind ...
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Problem in Rick Miranda - check that map is unramified

Let $U$ be the affine curve defined by $x^2=3+10t^4+3t^8$ and $V$ the curve defined by $w^2=z^6-1$. Both are smooth. Let $F:U\to V$ be defined by $(x,y)\mapsto (z,w)= (\frac{1+t^2}{1-t^2},\frac{2tx}{...
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Inertia and decomposition groups in composite extension

Let $L$ and $M$ be two number fields. I know that $\renewcommand{\Gal}{\mathrm{Gal}}\Gal(LM/L)\cong \Gal(M/L\cap M)$ by restricting automorphisms. Fix a prime $p\in L\cap K$, and let $\frak{p_1}$ and $...
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Etale covers of $\mathbb{G}_{m,k}$ in char 0

Let $k$ be a field of characteristic 0. It seems it is well known that étale covers of $\mathbb{G}_{m,k}=\operatorname{Spec}(k[T^{\pm 1}])$ are in bijection with étale covers of $\operatorname{Spec}(k(...
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Compute extension and intertial degree of $\mathbb{Q}_3(\sqrt[4]{-3}, i, \xi_3, \sqrt[3]{2})/\mathbb{Q}_3(\xi_3, \sqrt[3]{2})$

I am trying to compute the extension degree and intertial degree (or ramification index) of the local field extension $\mathbb{Q}_3(\sqrt[4]{-3}, i, \xi_3, \sqrt[3]{2})/\mathbb{Q}_3(\xi_3, \sqrt[3]{2})...
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Ramification of $5$ in $\mathbb{Q}( \sqrt[5]{n})$

I need to study the ramification of $5$ in $K=\mathbb{Q}(\sqrt[5]{n})$. I know that $5$ ramifies in $\mathbb{Q}(\sqrt[5]{n})$ because $5$ divide the discriminant, my question is about the possible ...
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Are there any nontrivial unramified extensions between two cyclotomic fields?

Fix $m$ and let $H$ be the Hilbert class field of $\mathbb{Q}(\zeta_m)$. I'm trying to show that $H\cap \mathbb{Q}(\zeta_n)=\mathbb{Q}(\zeta_m)$ for any $n$ such that $m\mid n$. To do this, I think ...
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A formula for the norm residue symbol at $v$-primary element

I have been trying to solve the exercise 2.12 of the book "Algebraic number theory" published by Fröhlich and Cassels, but I fail to reach the conclusion. Let me sum up the statement Let $K$ be a ...
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When is a non-archimedean prime $v$ of a number field unramified in a Kummer extension?

I have been working on the exercises of the book "Algebraic number theory" published by Frölich and Cassels. In one exercise, the following statement is given as a fact: Let $K$ be a number field ...
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infiniteness of completely split primes?

Let $K ⊂ L$ number fields. Prove that there are infinitely many primes of K that split completely in $L$. My attempt: Using this question: Infinitely many primes in the ring of integers , we know ...
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show that $pO_K=(p,(1-\zeta_p)^{p-1}.$ [duplicate]

$\text{Totally Ramified primes in cyclotomic fields:}$ Suppose we have $$ \Bbb F_p[x]/\overline{\Phi(x)} \cong O_k/pO_K.$$ If $ \ \overline{\Phi(x)}=\bar x^{p-1}+\cdots+\bar x+1=\large \frac{\bar x^...
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On the existence of tamely ramified extension

I'm stuck dealing with the proof of the existence of tamely ramified extension. Here is the theorem: Let $K$ be a field complete with respect to a discrete absolute value, and let $E/K$ be a totally ...
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Action of $\operatorname{Gal}(K(X))$ on the normalization of $X$ in $K(X)^\text{sep}$

I'm reading this part of The Stacks Project regarding ramification theory. Right before defining decomposition and inertia groups for schemes there is the following passage: Let $X$ be a normal ...
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Induced residue field extension

Let $K$ be a local field with ring of integers $\mathcal{O}_K$ and residue field $\kappa_K$. Let $f(x)$ be a monic irreducible polynomial of degree $n$ in $\mathcal{O}_K[x]$. Let $L=K[x]/ (f(x))$. As ...
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What does prime-to-p-part mean?

What does prime-to-p-part mean ?
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Two different ways of presenting the ring of S-integers

Ok, first, here is my question: Let $O$ be a Dedekind domain, $K$ its quotient field and let $S$ be a finite set of prime ideals in $O$. Let $A:=\{x\in K: \forall\mathfrak{p}\not\in S\ (v_\mathfrak{...
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Splitting in cyclotomic field $Q(\zeta_5)$

My question is about the behavior of primes $p\equiv 2,3,4 \bmod 5$ in $Q(\zeta_5)$ if they are inert or splitting in 2 primes or 4 primes??
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Subfield contained in decomposition field

I've heard that this fact is "known" or "can be shown" but I am having trouble finding a proof of it myself. Take a Galois extension $E/K$ of number fields and let $q\in E$ lie over a prime $p\in K$...
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Product of ramification groups: Is $G_i= H_i H'_i$ in totally ramified case?

With reference to my older question Product of ramification groups: Is $G_i= H_i H'_i$?, I want to specify it in the totally ramified case: Let $L/K$ be a finite Galois totally ramified ...
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Values of Grössencharacter attached to CM elliptic curve

Let $E$ be an elliptic curve defined over a number field $L$, having CM by by the ring of integers $\mathcal{O}_K$ for $K$ quadratic imaginary. If $K \subseteq L$, then (as constructed in Silverman's ...
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Product of ramification groups: Is $G_i= H_i H'_i$?

Let $L/K$ be a finite Galois extension of local fields with Galois group $G$. Suppose we have two linear disjoint Galois subextensions $E/K$ and $E'/K$ of $L/K$ with $EE'=L$. Let $H=G(L/E)$ and $H'=G(...
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How does this imply that $O_L/\mathfrak{p} O_L$ has non-zero nilpotent element?

Let $O_K$ be the ring of integers of an algebraic number field $K$ and let $\mathfrak{p}$ be a prime in $O_K$. For a field extension $L/K$, we consider the ring of integer $O_L$ in $L$ and the ideal $\...
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Inertia group of function field

Let $C(T)$ be a function field in the variable $T$ over an algebraically closed field $C$ of characteristic $0$. Consider L as the splitting field for the following polynomial: $F(X,T) = f(X)-T$ ...
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Field extension unramified at every prime over Q is unramified

This might be a weird question but I am quite confused about how unramified extensions and unramified primes are connected. The screenshot is a part of the proof of Kronecker-Weber and it basically ...
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About ramification groups

Let $K$ be a field complete with respect to a valuation $v_K$ on it. Let $K<L$ be a finite Galois extension that is totally ramified, and let $G$ be its Galois group. Let $A_K$ be the ring of $v_K$ ...
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Finite Unramified Galois Extension

I'm wondering whether finite unramified Galois extensions of p-adic field number fields (i.e. extensions of $\mathbb{Q}_p$ for some prime $p$) are cyclic? The absolute Galois group is isomorphic to ...
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Maximal Unramified Extension of a Galois Extension

I am reading Neukirchs ANT and I'm having some difficulties with abstract Galois theory and abstract valuation theory. I know that my question is quite specific and requires more information than what ...