Questions tagged [ramification]

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

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ramification group of infinite place of function field [closed]

For any subgroup G of Aut(F/Fq), let FG be the fixed subfield of F with respect to G, FG = {y ∈ F : σ(y) = y for all σ ∈ G}. The i-th ramification group Gi(P) of P|Q for each i > −1 is defined by ...
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Example of degree $n$ ramified, but not totally ramified extension

I'm looking for an example of degree $n$ ramified but not totally ramified example over $\Bbb Q_p$. I can find degree $n$ ramified extension, for example, $\Bbb Q_p({p^{1/n}})/\Bbb Q_p$. $p^{1/n}$'s ...
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Sufficient condition of $K'/K$ is ramified extension

Let $K$ be a local field and $K'$ be it's finite extension. And there exists $a∈K'$ such that $v(a)$ is not integer. Then, ramification index $e$ of $K'/K$ is at least 2, in other words, $K'/K$ is ...
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ramification index of local field is more than 2

Let $K$ be a local field and $K'$ be it's finite extension. And there exists $a∈K'$ such that $v(a)$ is not integer. Then, ramification index $e$ of $K'/K$ is at least 2, in other words, $K'/K$ is ...
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Ramifies as well as split

Can somebody give an example of a finite Galois extension of $\mathbb{Q}$ where a rational prime $p$ ramifies ( some (equivalently every) prime lying over $p$ has ramification index $> 1$) as well ...
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Prove any semistandard Ehrhart form has at least one totally ramified subform

I'm self-studying finite ramification theory, and I found the following problem: Let $\Phi$ be a semistandard Ehrhart form (i.e. its entries are weakly increasing). Prove that $\Phi$ has at least one ...
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A confusion on the definition of ramification group

I'm reading A Brief Guide to Algebraic Number Theory by Swinnerton-Dyer, and I'm stuck with the following sentence in page 28. I can't see why this implies the property of $\sigma$ doesn't depend on ...
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Unramified subextensions of $\mathbf{Q}(\alpha,\sqrt{-23})$

Let $\alpha$ be a root of the polynomial $f=X^3-X-1$. The following exercise should guide me through the standard example of a Hilbert class field. I showed that the class group of $\mathbf{Q}(\sqrt{-...
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Ramification question in compositum of cyclotomic and degree 5 extension.

I am reviewing some past exam questions and I have a problem solving the following example: Let $F = \mathbb{Q}(\zeta_5, \sqrt[5]{75})$, a field of degree 20 over $\mathbb{Q}$. Determine the ...
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Can inseparable elements “appear” in the residue field of the Galois closure of a field extension with separable residue field extension?

I am studying these notes and I am trying to generalize a bit the setting of the Section 3, because there doesn’t seem to be a fundamental reason to only study $p$-adic fields. So all the fields ...
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Quartic unramified extension of $\mathbb{Q}_p$ and existence of generators of the form $\sqrt[4]{a}$ for an integer $a$

Let $K=\mathbb{Q}_p$ where $p$ is a prime number such that $p-1$ is divisible by $4$. Furthermore, let $F/K$ be the unramified extension of degree $4$. Question: Does $F/K$ always have a generator of ...
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the example of field extension over $\mathbb{Q}_p$ such that the Galois group is not solvable [closed]

I would like to know the example of field extension over $\mathbb{Q}_p$ such that the Galois group is not solvable. It is well known that totally ramified extension of local fields corresponds with ...
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Terminology of splitting of prime ideals: How to say that the prime lying under is unramified?

Let $K$ be a number field, and let $\frak{P}$ be a nonzero prime ideal of the ring of integers of $K$. Is there a less cumbersome way to say "the prime number lying under $\mathfrak{P}$ is ...
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example of non abelian totally ramified extension

I would like to know the example of totally ramified extension which is not an abelian extension. It is well known that the Galois group of totally ramified extension is solvable. I studied the proof ...
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Odd subfield of cyclotomic field with 2-ramification

I have a tower of fields $\mathbb{Q} \subset L \subset \mathbb{Q}(\zeta_{m})$ with the following hypothesis. $[L:\mathbb{Q}]$ is odd, $2$ is the only rational prime that ramifies in $L$. I want to ...
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Discriminant of odd cyclic Galois extension is not a power of 2

Is there a way to prove that, given a Galois cyclic extension $\mathbb{Q} \subset F$ with odd order, there exists a prime $p \neq 2$ such that $p|\Delta_{F}$ ? I'm actually trying to prove that the ...
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How to show that this $V_4-$ extension ramifies at the infinite prime?

A result by Boston and Markin states that the minimum number of primes that ramify in a $G-$extension of $\mathbb{Q}$ is at least $d(G^{ab})$, the minimum number of generators of the abelianisation $G/...
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Contraction of square of ideal is square of contraction - unramified primes of splitting field

The question is about the answer to this question. Basically we have a ring extension $R\subset S$ where $S/R$ is finite (quotient as abelian groups), $S$ is a Dedekind domain and an ideal $P$ of $S$. ...
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Relationship between the generators of two different cyclic and totally ramified extensions of $\mathbb{Q}_p$

Let $K=\mathbb{Q}_p$ and $L/K$ be a cyclic and totally ramified extension of degree $n$, generated by an element $\alpha$ (i.e. $L=K(\alpha)$). Let $L'/K$ be another cyclic and totally ramified ...
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A problem related to localization

Let $A$ be integrally closed (not necessary be Dedekind), $K=\operatorname{Frac}(A),L/K$ Galois, $B$ is the integral closure of $A$ in $L,p$ is a maximal ideal in $A$. We know that the galois group $G$...
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torsion element of group associated to formal group

Let $R$ be complete local ring $M$ be the maximal ideal of $R$ $F$ be a formal group defined over $R$, with group law $F(X,Y)$. According Silverman's book 'the arithmetic of elliptic curves', example ...
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Ramification indices and residue class degrees of $\mathfrak{O}_K$ where $K=Q[\alpha]$, $f(\alpha)=0$, and $f(x)=x^3-x-1$.

I already know that $\alpha^3-\alpha-1=0$ implies that $\{1,\alpha, \alpha^2 \}$ creates an integral basis for $\mathbb{A} \cap \mathbb{Q}[\alpha]$. I'd like to try to use Dedekind's theorem in some ...
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Is it always possible to construct an extension of $\mathbb{Q}_p$ as a totally ramified extension first and then an unramified extension after?

Let $K = \mathbb{Q}_p$, and $L/K$ be a finite Galois extension. Question: Is it possible to find a totally ramified extension $L'/K$ such that $L/L'$ is unramified? I know that it is always possible ...
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Marcus, Number Fields, Problem 19(b), Chapter 4

It would be really helpful if somebody could check my argument! Marcus suggests something else but I (think) I found another proof. So here goes: First of all let's set the stage: Let $L/K$ be a ...
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branch point of elliptic curve

Sorry for my bad English. Let $X$ be elliptic curve over algebraic closed field $k$, and $P_0\in X$. By Hartshorne's Algebraic geometry, linear system $|2P_0|$ define morphism $f:X\to \mathbb{P}^1$...
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Prime ramifies in L/K iff it ramifies in the Galois closure of L/K

I want to show that $p$ is unramified in a field extension $L/K$ if and only if it is unramified in the Galois closure $N/K$. This is what I have so far: if $p$ is unramified in $N/K$, then because ...
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How we can know the ramification ideals geometrically?

Let $L/\mathbb{Q}$ be a finite Galois extension of degree n, let $\mathcal{O}_{L}$ be the ring of integers of $L$, By Dedekind lemma we have that $\mathfrak{p}=\mathfrak{b}_{1}^{e}...\mathbb{b}_{g}^{e}...
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Definition of inseparable morphism over non projective curve

Sorry for my bad English. I’m trouble with a proof of Hartshorne of IV.proposition 3.8. But I don’t know definition of inseparable over non projective curve $X-R$. If there is explanation or good ...
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From general ramification theory to ramification indexes of morphisms of elliptic curves

The reference for what I'll say here is Silverman's The arithmetic of elliptic curves, ch. 2, § 2. Let $C_1, C_2$ be two elliptic curves, $\Phi : C_1 \to C_2$ a non trivial morphism of algebraic ...
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Ramification in subfield lattice of a cyclotomic field

Fix a positive integer $n=p_1^{a_1}\cdots p_k^{a_k}$. Consider the cyclotomic field $K=\Bbb Q(\zeta_n)$. I know that only $p_1,\ldots,p_k$ are ramified in $K/\Bbb Q$, and that the ramification index ...
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Inertia degree of primes in p-adic extensions

I'm reading through some number theory and ran across a theorem where the proofs referenced were incomprehensible to me, and I was hoping there might be a simpler proof than slogging through another $\...
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Galois group of the compositum of a non-totally-ramified and a the unramified extension of $\mathbb{Q}_p$ of prime power degree

Let $K=\mathbb{Q}_p$ and $L,F$ be extensions of $K$ such that $[L:K] = [F:K]$ is a prime power, $L/K$ is not totally ramified, $F/K$ is unramified, $L/K$ is cyclic (and $F/K$ too which is implied by ...
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Why are the extensions of valuations in a Galois extension conjugate under the action of the Galois group?

I'm reading the following paper by Hansen about Galois Coverings of Curves: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.9776&rep=rep1&type=pdf (see page 37) Unfortunately, I'm ...
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Ramification and algebraic closure

Let $k$ be a field and $f:C\rightarrow B$ a morphism of smooth curves. Let $C'$, $B'$ and $f'$ be the extensions of $C$, $B$, and $f$ to $\overline{k}$. Let $c\in C$, $b=f(c)$, $c'\in C'$ above $c$ ...
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When can we say that $e(LK/L)\leq e(K/\mathbb{Q}_p)$?

Fix a prime $p$ and let $L$ and $K$ be two finite extensions of $\mathbb{Q}_p$ with ramification degrees $e_L$ and $e_K$, respectively. Let $e'$ and $e''$ be the ramification degrees of $LK/L$ and $LK/...
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If $\phi:X\to Y$ is a morphism of curves, why $\sum_{f(P)=Q}e_P=\deg\phi$ for all $Q\in Y$?

Let $X,Y$ be projective varieties over an algebraically closed field of characteristic zero. If $\phi:X\to Y$ is a surjective morphism, which induces a finite field extension $f^*:k(Y)\hookrightarrow ...
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Lower for the inertial degree $f$ in cyclotomic field

Let $\mathbb{Q(\zeta)}/\mathbb{Q}$ be a galois extension of degree $p-1$ where $\zeta=e^{\frac{2 \pi i}{p}} $ Let $\mathfrak{b}$ be an ideal above $q$ Where q is a prime in $\mathbb{Z}$ different from ...
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Inertia subgroup of finite extension over $\mathbb{Q}_p$ whose ramification index is not divisible by $p$

Let $K$ be an extension of $\mathbb{Q}_p$ and let $L/K$ be a finite extension with $p \nmid e$ where $e = e(L/K)$ is the ramification index of $L/K$. Let $I=I(L/K)$ be the intertia subgroup of $L/K$. ...
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On the factorization of prime ideals of $\mathbb{Z}$ in $\mathbb{Z}[\zeta]$

Let $\mathbb{Q(\zeta)}/\mathbb{Q}$ be a galois extension of degree $p-1$ where $\zeta=e^{\frac{2 \pi i}{p}} $ and let $G=\operatorname{Gal}(\mathbb{Q(\zeta)}/\mathbb{Q})$ be its Galois group. Suppose ...
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Analogy between $\operatorname{Spec} \Bbb R[y] \to \operatorname{Spec} \Bbb R[y^2]$ and $\operatorname{Spec} \Bbb Z[i] \to \operatorname{Spec} \Bbb Z$

Consider two (covering) flat morphisms: $$f: \operatorname{Spec} \mathbb{R}[y] \to \operatorname{Spec} \mathbb{R}[x], y^2=x$$ and $$g: \operatorname{Spec} \mathbb{Z}[i] \to \operatorname{Spec} \mathbb{...
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Galois representation being unramified is Galois local

Let $K$ be number field and $\rho:G_K\rightarrow \text{Gl}(V)$ a Galois representation. Let $\nu$ be a place of $K$ (non-archimedean if it helps/is necessary). We say that $\rho$ is unramified at $\nu$...
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What does 'Let $F: X \to Y$ be a holomorphic map defined at $p \in X$, which is not constant' mean?

I refer to Chapter II.4 of Rick Miranda - Algebraic curves and Riemann surfaces. There's this statement Let $F: X \to Y$ be a holomorphic map defined at $p \in X$, which is not constant. Here, $X$ ...
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Whether or not constant multiplicity one implies chart map

I refer to the exercises of Chapter II.4 of Rick Miranda - Algebraic curves and Riemann surfaces. Question: Can Exercise II.4E help answer the 2nd part of Exercise II.4A (about converse)? Guess: I ...
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If $F$ has a ramification point, then is $F$ necessarily not injective?

I thought of this question based on my other question here. I understand that for a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $X$ and $Y$, both of which are connected but not ...
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Finding a generator of a cyclic and totally ramified extension by using a generator of an unramified extension of the greater field of the same degree

Let $K$ be an extension of $\mathbb{Q}_p$ which contains a primitive $n$-th root unity. Also, assume that $p$ does not divide $n$. Let $L/K$ be a cyclic and totally ramified extension of degree $n$, ...
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Example of number field with certain conditions on ramification index and degree

I am looking for a number field with degree $n$ over $\mathbb{Q}$ and with a ramified prime $p$ with ramification index $e$ such that $\textrm{gcd}(n, p-1) = 1$ and $\textrm{gcd}(e, p-1)>1$. I ...
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Step in proof of Hasse-Arf theorem on ramification groups

This question concerns Yoshida's proof of the Hasse-Arf theorem in https://arxiv.org/abs/math/0606108 (page 16). For a totally ramified extension $K^\prime/K$ of local fields define the ramification ...
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60 views

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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