Questions tagged [ramification]

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

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abelian extension of local field is unramified

Why is this result true : $K$ being a local field with algebraicaly closed residue field then an abelian extension of $K$ (let's say $L$) is totaly ramified! I didn't find any reference in all the ...
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Unique unramified ideal implies that the ramification index is equal to the degree of field extension in a galois extension

Given a Galois extension $K \supseteq \mathbb{Q} $, prove that if there is only one unramified prime number $p$ over $K$ then there is only one prime ideal $\mathfrak{p} \subseteq O_K$ containing $p$ ...
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Relation between inertia group in character theory and commutative algebra

When studying character theory (specifically, of normal subgroups), one comes across the concept of the inertia group. If $N \unlhd G$, where $G$ is a finite group, then, $G$ acts on $\operatorname{...
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Splits completely of a prime ideal

Suppose that $K$ is a number field and $\mathfrak p$ is a prime ideal non-zero. In general always exists a finite extension of $L$ of $K$ such that $\mathfrak p$ is ramified, for example $L=K(\sqrt f)$...
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Transitive Galois action on the set of prime ideals

Suppose that $K$ is a number field and $\mathfrak p$ is a prime ideal non-zero. Suppose that for each $n\in\mathbb N$ there exist a finite Galois extension $K\subseteq L$ such that $$\mathfrak p=(\...
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Powers of a uniformizer in a totally ramified extension of $\mathbb{Q}_p$

Consider the cyclotomic extension $\mathbb{Q}_p(\zeta_{p^n})/\mathbb{Q}$. This is a totally ramified extension of degree $\phi(p^n) = p^{n-1}(p-1)$ ($\phi$ is the Euler totient) and a particular ...
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Composition unramified for every extension

Let $K$ be a number field and $S$ be a finite set of primes. Is it possible to construct a finite extension $M$ of $K$ such that $LM/M$ is unramified at (the primes above) $S$ for all degree $n$ ...
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Geometric interpretation of Lefschetz number for local fields

I have the following problem. Let $L/K$ be a finite galois extension of local fields with Galois group $G$. For nontrivial $g\in G$ define Lefschetz number $i_{L/K}(g):= \min\limits_{x\in \mathcal{O}...
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Hasse Symbol for ramified primes

My Reference is https://virtualmath1.stanford.edu/~conrad/249BW09Page/handouts/cfthistory.pdf In chapter 7, page 17, Keith says For example, $(\alpha, L/K)_v$ lies in the common decomposition group $...
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Finite group acting on projective line effectively and holomorphically

Recently I'm working on Exercise I of Problem III.3 of "Algebraic curves and Riemann surfaces". Define holomorphic and effective actions of $A_4,S_4$ and $A_5$ on the projective line such ...
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Does there exists a quadratic extension $L$ of $K$, in which $p_1,p_2,\ldots,p_n$ ramifies in $L/K$?

Let $K$ be a number field. For arbitrarily fixed positive integer $n$, fix $p_1,p_2,\ldots,p_n$ be a prime elements of ring of integers of $K$. Does there exists a quadratic extension $L$ of $K$, in ...
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Ramification theory, What is the relation between $K_S$ and $L_S$?

Let $L/K$ be a finite extension of number field. Let $S$ be a set of places of $K$ containing Archimedean places. Let write $K_S$ for the largest subfield of $\overline{K}$ containing $K$ that is ...
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Why $ev_p$ is normalized, where $v_p$ is the $p$-adic valuation of $\mathbb{Q}_p$ being extended to $K$? ( Neukirch's ANT book, (5.5) Proposition)

I am reading the Neukirch's Algebraic number theory, p.138, proof of the II-(5.5) proposition and stuck at some point. (5.5) Proposition. Let $K|\mathbb{Q}_p$ be a $\mathfrak{p}$-adic number field ...
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What is the ramification group of a curve at a point

I'm reading the book "Weil conjectures, perverse sheaves and l-adic fourier transform" I can't understand the following lemma:  where $X_0$ is a smooth curve over $\kappa=\mathbb{F}_q$, $X=...
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Prime of order is regular iff its decomposition in the normalization is trivial.

It's from a statement in Algebraic Number Theory by Neukirch, page 92. Example 5 "One can show..." Let ${o}$ be a one-dimensional noetherian integral domain and $\tilde{o}$ be its ...
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How to find vanishing order of coefficients of Newton polynomial?

I'm given the curve $y^3=x^3-1$ and I want to find the genus. I have a ramified cover $\pi: Y \to \mathbb{P}^1$, where $Y$ is given by the curve. I know that $1, \omega, \omega^2$ are the $x$ values ...
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Example of unramified covering map of degree d

In my self-study of Lec.10 of Belyi Maps and Dessins d'Enfants, I came across the following statement [Right after Remark 7] Let $X$ and $Y$ be Riemann surfaces and $\pi: Y \rightarrow X$ be an ...
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Infinite primes in number fields

David Cox's Primes of the form $x^2+ny^2$ defines infinite primes as [Infinite primes] are determined by the embeddings of $K$ into $\mathbb{C}$. A real infinite prime is an embedding $\sigma: K \to \...
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Is the splitting field of $x^3+2x^2-x-3$ a totally ramified extension of $\mathbb Q_5$?

Consider the polynomial $f(x)=x^3+2x^2-x-3 \in \mathbb Q_5[x]$. Is the field $K:=\mathbb Q_5[x]/(f(x))$ totally ramified ? Through Newton polygon argument, I see $f$ is irreducible in $\mathbb Q_5$. ...
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Galois extension of $\mathbb{Q}_p$ with Galois group $\mathbb{Z}/2\mathbb{Z}$

Denote by $\mathbb{Q}_p$ the $p$-adic completion of the integers, i.e. with respect to the valuation $|x|=p^{-ord_p(x)}$. My question is, how do we find, and classify explicitly, all the field ...
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Ramification Index 2 in Galois closure of field with squarefree discriminant

This is about Exercise 7 from here. Let $K$ be a number field of degree $n$ with Galois group $S_n$ whose discriminant$D$ is squarefree. Prove that the Galois closure of $K$ is unramified over all ...
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Step in Yoshida's proof of Hasse-Arf theorem

This question concerns Yoshida's proof of the Hasse-Arf theorem in the local class field theory in https://arxiv.org/abs/math/0606108 (page 16). For a totally ramified extension $K′/K$ of local fields,...
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About (wild) ramification of an abelian extension and its conductor

Let $K/\mathbb{Q}$ be an abelian extension, and define the conductor of $K$ as the smallest integer $n$ such that $K\subset \mathbb{Q}(\zeta_n)$. How does one show the following two facts: The ...
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If there exists a prime $l$ such that $l$ spilts in $K$, why does that imply there exists a place of $K$, which satisfies $K_v \cong \Bbb{Q}_l$?

Let $K$ be a number field. If there exists a prime $l$ such that $l$ spilts in $K$, why does that imply there exists a place of $K$, which satisfies $K_v \cong \Bbb{Q}_l$ ? Here, $K_v$ denotes ...
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Surjectivity of norm map on principal units in a totally ramified extension of local fields

I have a question about a statement in Chapter V, section 3 of Serre’s book on local fields. Let $\ell$ be a prime number, and let $L/K$ be a cyclic Galois extension of local fields of degree $\ell$, ...
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Example of rational jumps in upper ramification filtration

Can anyone point to me or write down an explicit example of a non-integer jump of the higher ramification filtration in positive characteristic and the corresponding equations of the intermediate ...
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What is an Archimedean prime?

I need to learn some infinite ramification theory and I am stuck with understanding it. I understand that we consider the order of the inertia group as the ramification index, and if the inertia group ...
Izzy Garcia's user avatar
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$\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n \notin \Bbb{Q}(ζ_{11})$ for all positive integer $n$

I want to prove $\left(i\sqrt {\frac{21 \ + \ 5\sqrt{5}}{2}}\right)^n$ does not lie in $\Bbb{Q}(ζ_{11})$ for all positive integer $n$. This problem arises from arithmetic geometry, but this problem ...
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If $Q \vartriangleleft \Bbb Z[i]$ lies over $(p) \vartriangleleft \Bbb Z$ where $p\in \Bbb Z\setminus\{2\}$, then $e(Q|p) = 1$.

Let $R = \Bbb Z$, $S = \Bbb Z[i]$, and $p \in R\setminus\{2\}$. If the ideal $Q \vartriangleleft S$ lies over $(p) \vartriangleleft R$, then $e(Q|p) = 1$. The primes (non-zero prime ideals) in $S$ ...
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Construct an extension of a number field of given degree where a given set of prime splits completely

I know that discriminant tells exactly which prime ramify in an extension, and it helps to construct extensions where a certain set of primes will ramify. But I don't know how to construct extensions ...
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Maximal inert extension

I am wondering is there a theory on maximal inert fields? Consider a Galois extension of number fields $K\subset L$ with the Galois group $G=G(L/K)$ and rings of integers $R_K\subset R_L$. Let $p\...
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Proof about the closed integral $\mathbb{Z}_p[\zeta_p]$

Assuming $\zeta_p$ is a root of unit, I need to show that $\cal{O}$$(\mathbb{Q}_p(\zeta_p))=\mathbb{Z}_p [\zeta_p]$. Where $\cal{O}$$(F)=${$x\in F:|x|_p\le 1$} and $\mathbb{Z}_p$ is the ring of p-adic ...
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Decomposition of extensions of local fields into an unramified and totally ramified one

Take a finite separable extension $L/K$ of non-archimedean local fields. For example, $K = \mathbb{Q}_p$ and $L$ some finite extension. We know that we can decompose $L/K$ into a tower $L/K_0/K$ where ...
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Totally tamely ramified compositum

Let $E/F/\mathbb Q_p$ be a tower of finite extensions of $p$-adic fields, and fix an algebraic closure. Let $\pi\in F$ be a uniformiser, let $p\nmid s$, and let $\alpha$ be a root of $X^s-\pi$. Then $...
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For a finite extension $K/\mathbb{Q}_2$, the extension $K(\sqrt{-1})/K$ is always totally ramified

Let $K$ be a finite extension of the $2$-adic numbers $\mathbb{Q}_2$, and suppose that $-1$ is not square in $K$. Write $K(i)$ for the quadratic extension $K[X]/(X^2 + 1)$, where $i^2 = -1$. Is it ...
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Why Does (3) Completely Ramify in $\mathbb{Q}(\omega, \sqrt[3]{2})$?

I'm working through this notes on Algebraic Number Theory. In section 2.6 they claim (3) is completely ramified over the larger field: If we know that the ring of integers of $\mathbb{Q}(\omega, \sqrt[...
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Hilbert class field of $ \Bbb{Q}( \sqrt{-5})$

I want to prove $ \Bbb{Q}( \sqrt{-1},\sqrt{-5})$ is Hilbert class field of $ \Bbb{Q}( \sqrt{-5})$. Let $H$ be Hilbert class field of $ \Bbb{Q}( \sqrt{-5})$. I know already from elliptic curve theory ...
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Good references for Kronecker-Weber Theorem? [closed]

I am looking for references to self-learn the proof of the Kronecker-Weber theorem in a way which takes the least prerequisites and can be understood easily and nicely. Please give your suggestions ...
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Consequences of definitions in Local fields

I am currently taking a course on Local Fields and have some tiny questions I cannot seem to wrap my head around: Question 1: If $L/K$ is an extension of complete discretely values fields with ...
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Why restrict the domain of homomorphism to decomposition group?

Let $L/K$ be a number field abelian extension. Let fix a prime ideal $p$ of $K$.  Then there is surjective homomorphism from decomposition group $D_p$ to corresponding residue field extension.  My ...
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Number of prime ideal of $ \Bbb{Z}[ \sqrt{-5}]$ above $(2)$

I want to prove number of prime ideal of $ \Bbb{Z}[ \sqrt{-5}]$ above $(2)$ is 1. My try: From ramification theory, (number of prime ideal above $(2)$)=[$\Bbb{Q}[ \sqrt{-5}:\Bbb{Q}]/$(order of ...
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Compositum of infinitely many unramified extensions is unramified

The fact that if I have two extensions of number fields, say $L_1 / K$ and $L_2 / K$, unramified at a discrete valuation $v$, then the compositum $L_1L_2 / K$ is unramified at $v$ can be proven ...
stillconfused's user avatar
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Homomorphism from inertia group to $(\mathcal{O}_L/Q)^*$

(This is exercise 21.b in Chapter 4 of Marcus' Number Fields) Let $L/K$ be a normal extension and let $Q$ be a prime ideal of $\mathcal{O}_L$. Fixing $\pi \in Q - Q^2$ and considering an automorphism $...
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To find an element of galois group which sends one prime ideal above $p$ to another prime ideal above $p$

Let $L/K$ be finite galois number field extension and $p$ be prime of (ring of integers of) $K$. Let $G=Gal(L/K)$. It is well known that $G$ acts transitively on the set of all prime ideals of $L$ ...
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$L/K$ is unramified extension implies corresponding local extension $L_P/K_p$ is unramified?

Let $K$ be a number field and $L/K$ be finite galois extension. Let $p$ be a prime ideal of ring of integers of $K$. Let $P$ be a prime ideal above $p$. Let's think about corresponding local extension ...
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Corresponding local extension does not depend on choice of prime ideal above?

Let $L/K$ be extension of number field. Let $p$ be a prime ideal of $K$. Let $P$ be a prime ideal above $p$. Then we can define local field extension $L_P/K_p$,where $L_P$ and $K_p$ is completion at $...
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Why order of decomposition group=$[L:K]/$(number of prime ideal above $ \mathfrak p$)

Let $L/K$ be a finite galois extension of number field. Let $\mathfrak p$ be prime ideal of $K$ .$$D_{\mathfrak p} = \{ \sigma \in G : \sigma(\mathfrak p) = \mathfrak p\}$$ be decomposition group. ...
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Relative degree 1 prime ideals

I am reading Algebraic Number Fields by Gerald Janusz. Let $K$ be a number field with number ring $R$. In chapter IV prop 4.3, he tries to apply zeta functions to prove there are infinitely many prime ...
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Totally ramified extension of global field in terms of local field case

Let $K$ be a number field of class number $1$(in other words, it's ring of integers $O_K$ is PID. Let $p$ be a nonzero prime ideal of $O_K$. $L/K$ is an extension with $n=[L:K]$, we say that $L$ is ...
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Galoisian and abelian covering of elliptic curve ramified only above one point

As the title suggest, I have a question about abelian (thus Galois) cover of an elliptic curve ramified only above one point. Actually, I'm pretty confused if it exists or not. To make things clear, I ...
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