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Questions tagged [ramification]

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

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For what fields does this morphism from an elliptic curve to the projective line ramify at infinity?

Let $k$ be a field. Consider the curve $X := V_+(X_1^2 X_2 + X_1 X_2^2- X_0^3-X_0^2X_2) \subseteq \mathbb{P}^2_k = \text{Proj}(k[X_0,X_1,X_2])$. Consider the morphism given on functions fields by $k(...
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ramification index

what is the ramification index and inertia degree of the extension $\mathbb{Q}_2$(i)/$\mathbb{Q}_2$. From definitions, index e = e(w|v) = (w(L*) : v(K*)) is called the ramification index of the ...
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some notation in field extension

I know $\mathbb{F}_3$ = $\mathbb{Z}$/3$\mathbb{Z}$ = {0,1,2}. But what does $\mathbb{F}_3$ ((X)) mean? And how can we find a totally tamely ramified extension and unramified extension of it ...
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Dedekind theorem of ramification in cyclotomic fields

Let $\Bbb Q(w)$ denote the $n$-th cyclotomic field then$\Bbb Z[w]$ is its ring of integers, $d$ denotes the discriminant of $\Bbb Q(w)$, and $p\mid d$ then $p\Bbb Z$ must ramify in $\Bbb Q(w)$. In ...
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Unramified extensions and base change

Let $K'/K$ be a finite extension of local discrete valued fields, and let $E/K$ be any extension of local discrete valued fields. Assume that $E\otimes_KK'$ is a field, and that $E\otimes_KK'/E$ is ...
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Understanding the $p$-part of the discriminant of a totally real number field with a single prime above $p$

Let $K$ be a totally real Galois number field, and suppose there is only one prime above $p$, with ramification index $\leq p-1$. If $K_p$ is the completion of $K$ at the prime above $p$, the claim ...
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Why is $\mathbb{C}$ over $\mathbb{R}$ considered ramified?

For a number field $K/\mathbb{Q}$, we say that a finite place of $Q$ is ramified if there exists a valuation $v_{p_i}$ in $K$ lying over $v_p$ such that it is ramified in the sense of the associated ...
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Is the Inertia quotiented by the first group of ramification $I/G_1\simeq (\mathcal{O}_L/P)^*$?

Consider $K$ number field, and $L/K$ a finite Galois extension with $G$ its Galois group and $P$ a prime of $\mathcal{O}_L$ above $p$ prime in $K$. Moreover suppose that the characteristic of the ...
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Galois Groups in Ramification Theory

I had a slight confusion about Galois groups over a base field which is complete with respect to a discrete valuation. We know that there are irreducible polynomials such as $X^3+X^2+2X-8$ where ...
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Computing degrees and ramification indices of some extensions of $\mathbb{Q}_2$

Let $K=\mathbb{Q}_2$ and $F = K(\zeta_3,\alpha)$ where $\zeta$ is a primitive third root of unity and $\alpha$ is a cubic root of $2$, i.e. $\alpha^3 = 2$.Let $K_1 = K(\zeta_3)$, $K_2 = K(\alpha)$ and ...
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Proof of a Lemma for a local field extension with certain properties

The following result is from "Euler Factors determine local Weil Representations" by Tim and Vladimir Dokchitser: Lemma 1: Let $F/K$ be a cyclic extension of degree $n$ and ramification degree $e$. ...
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Ramified Field Extensions

Let $k$ be a field of $char(k)=0$ und we consider an field exension $L/k$ with $[L:k]=n$. Set $M:= L((t^{1/n}))$ and $F:= k((t))$. I'm looking for a proof of following two statements: 1) If the ...
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Construction of cyclic local field extensions of arbitrary degree and ramification index

Let $K$ be a local field. Let $n$ be an arbitrary natural number and $e$ be any divisor of $n$. Question Does there exist an extension $L/K$ with the following properties? $L/K$ is a cyclic ...
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Image of a character remains the same when restricting to a totally ramified extension

Problem I want to prove: Let $\chi: G_K \to \mathbb{C}^*$ be an unramified character and let $L/K$ be a cyclic totally ramified extension. Then $\chi(G_K)=\chi(G_L)$. All I managed to do was ...
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Relation between local and global inertia/ramification degrees

Let $K/\mathbb{Q}$ be a number field and suppose a prime $p\in\mathbb{Z}$ factors in $\mathcal{O}_K$ as $\prod_{i=1}^r \mathfrak{p}_i^{e_i}$. From algebraic number theory, we have the identity $$ [K:\...
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Unramified subextension of the Galois closure of a totaly ramified $p$-adic field

Let $L/L'/K/\mathbb{Q}_p$ be a tower of finite field extensions such that $L'/K$ is totally ramified and $L/K$ is its Galois closure. We suppose that $L/L'$ is unramified. Let $M/K$ be a Galois ...
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Characterization of finite cyclic totally ramified extension of local fields with prime power degree

Definition Let $G_K$ be the absolute Galois group of a local field $K$. We will call a group homomorphism $\chi: G_K \to \mathbb{C}^*$ with finite image a character on $K$. Since every finite ...
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Inertia coprime to degree implies Inertia cyclic?

Let $K$ be a number field and let $\mathfrak{p}$ be a prime of $K$ co-prime to 2. Let $L/K$ be a Galois extension of degree a power of 2. Let $I$ denote the inertia group for $\mathfrak{p}$ relative ...
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109 views

Why can we always restrict an Galois representation so that it becomes unramified?

Let $K$ be a local field and $\rho: G_K \to \operatorname{GL}_n(\mathbb{C})$ be a Galois representation where $G_K$ denotes the absolute Galois group of $K$. We call a Galois representation $\rho$ ...
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How can one check if two totally ramified extensions of the same degree are equal?

Let $F/K$ and $L/K$ be finite and totally ramified extensions of local fields whose degree are equal. Question: How can we check if $F=L$ or not? My thoughts and attempts: If we look at $K=\mathbb{...
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Can one write a finite extension of local fields as a compositum of fields whose degrees are prime powers?

Let $F/K$ be a finite extension of local fields of degree $n$. Question: Does there exist intermediate fields $F/K_i/K$ such that the degree of $K_i/K$ is a prime power and $F$ is the compositum of $...
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Is an unramified extension of $Q_p$ complete w.r.t. the extension of the p-adic norm?

I'm studying for a course in number theory and I have seen that: $\Bbb Q_p$ is complete w.r.t. the $p$-adic norm $\Bbb Q_p^{unram}$, the union of all unramified extensions of $\Bbb Q_p$, is not ...
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Ramification degree of field extension of the 5-adic field obtained by adjoining a primitive third root of unity

Let $K = \mathbb{Q}_5$ and $K' = \mathbb{Q}_5(\xi_3)$ where $\xi_3$ is a primitive third root of unity, i.e. $\xi_3, \xi_3^2 \neq 1$ but $\xi_3^3=1$. The minimal polynomial of $\xi_3$ over $K$ is $x^2+...
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ramification at infinity of a Galois Extension.

Let $\mathbb F_q$ be a finite field, let $f\in \mathbb F_q[x]$, and let $t$ be trascendental over $\mathbb F_q$. Consider the splitting field $M$ of $f-t$ over $\mathbb F_q(t)$. Let $P_\infty$ be the ...
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Maximal extension of $\mathbb Q$ unramified outside a finite set of primes

I read the beginning of chapter 1 of Wiles paper on Modular elliptic curves and Fermat's Last Theorem. There it says: Let $p$ be an odd prime. Let $\Sigma$ be a finite set of primes including $p$ ...
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Characteristic of residue field in a Dedekind domain.

Let's consider a Dedekind domain $A$ with field of fractions $K$. Let $L$ be a finite Galois extension of $K$ and $B$ the integral closure of $A$ in $L$. Let $\mathfrak{p}\subset A$ be a non-zero ...
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A doubt on solvable groups and algebraic Number Theory

In Marcus book “Number Fields” I have this exercise: (page 124, number 25) Let $L$ be a normal extension of $K$ and suppose $K$ contains a prime which becomes a power of a prime in $L$. Prove that the ...
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87 views

Lagrange interpolation with multiplicities

I was wondering if it were possible to do Lagrange interpolation with multiplicities. Lagrange interpolation gives us a polynomial that obtains certain values on a given set of points. More precisely, ...
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Prime ramification on cyclotomic fields

Hello all i have a question on ramification over a cyclotomic field. Let $p$ be any prime in $\mathbb{Z}$ and $k$ be any integer $\geq 1$. Consider the cyclotomic field $\mathbb{Q}(\zeta_{p^k})$. Is ...
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Maximal Totally Ramified Extension

Hi, The lemma above is from Iwasawa's Local Class Field Theory. Could you explain, why M = F' ? Thank you
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algebraic curve's branchpoints

Following along with this example: Understanding Ramification Points makes me question some things about ramification, and in the context of this curve: $$c(x,y,e) = 4\,x\,y^4 + 2\,y^2 - e\,y - 1$$ (...
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Ramification Groups without valuations

Let $L,K$ be number fields, with $L/K$ a Galois extension, and $Q$ a prime ideal in $L$. Let $E$ be the inertia group, and $H=\{\sigma \in Gal(L/K) : \sigma(x) \equiv x \pmod {Q^2}\,\forall x\in O_L\}$...
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What is $[O_{L}:O]$ and $[O_{L}/(\pi):O/(\pi)]$?

I try to understand the following proof: Let L/K be a finite, Galois extension. If L/K is unramified, then there is a canonical isomorphism Gal(L/K) $\cong$ Gal($k_{L}$/k) where $k_{L}$ is the ...
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Ramification in $\mathbb{Q}$ adjoint square roots of prime numbers

Let $K = \mathbb{Q}(\sqrt{p_1}, ..., \sqrt{p_n})$, where the $p_i$'s are distinct prime numbers. Let $p$ be a prime such that $p \neq p_i$ for all $i$, and $p \neq 2$. Why is it true that $p$ doesn't ...
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Which primes are ramified?

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we ...
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Geometric interpretation of ramification of prime ideals.

I am trying to understand geometrically the ramification of primes in a finite separable field extension. Let $A$ be a Dedekind domain with fraction field $K$ and $L/K$ a finite separable field ...
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Splitting of primes in a Galois extension

Let $L$ be the splitting field of $x^3+2x+1$. I want to know how 59 splits in $L$. I calculated the discriminant of $\mathbb{Z}[\alpha]$ to be $-59$, (where $\alpha$ is a root of the polynomial), ...
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Ramification of primes in the ring of integers of a number field

Let $K$ be a number field and let $\mathcal{O}_K$ be its ring of integers. Since $\mathcal{O}_K$ is a Dedekind domain, every ideal has a unique factorisation into a product of prime ideals. Let $(p)$ ...
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Given a prime $\wp$, are there different primes $\wp_2$,…, $\wp_s$ such that $\wp \wp_2\cdots \wp_s$ is principal?

Let $A$ be a Dedekind domain. Let $\wp$ be a prime ideal in $A$. Can one find pair-wise different prime ideals $\wp_2$,..., $\wp_s$, also different from $\wp$, such that their product $\wp\wp_2 \cdots ...
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Higher Ramification Groups of Degree $p$ Cyclic Extensions.

For the person who is about to report this as a duplicate, "no, this isn't a duplicate". So I am trying to prove the following two (related) problems: Suppose $K$ contains a primitive $p$th root of ...
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Proving that $\sum_{v\in M_K, \,\,v\text{ archim.}} n_v=[K:\mathbb{Q}]$

Let $L/K$ be an extension of number fields, $M_K, M_L$ complete sets of representatives of places at $K$ and $L$, respectively. I'm familiar with the formula: $$\sum_{w\in M_L, \,\,w|v} n_w=[L:K]n_v\,...
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110 views

Finitely many unramified extensions with bounded degree

Let $K$ be a fixed number field, $d>0$ a fixed natural number and $S\subset \text{Spec}(O_K)$ a fixed finite subset of primes. I'm trying to prove the following statement: There are finitely ...
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Isomorphism of quotient of algebraic number rings

Assume $L,K$ are two number fields and $\mathcal{O}_L,\mathcal{O}_K$ are their algebraic number rings. Let $p_L (p_K)$ be a prime ideal for $\mathcal{O}_L$ $(\mathcal{O}_K)$ respectively, $n,m >1$ ...
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Ramification of primes in a number field's normal closure.

Let $K$ be a number field, $\mathcal{O}_{K}$ it's ring of integers. Suppose $p$ is a prime in $\mathbb{Z}$ such that d = disc($\mathcal{O}_{K}$) is exactly divisible by p$^{m}$ with $m$ odd. Prove ...
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$L/K$ unramified at $v$ and $\widetilde{K}$ intermediate field $\Rightarrow\, \widetilde{K}/K$ unramified at $v$

Let $K$ be a number field, $L$ a field with $K\subset L\subset\mathbb{C}$ and $L/K$ Galois (not necessarily finite). I'm trying to prove the following: Let $L/K$ be unramified at a finite place $v$...
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Classifying prime ideals of $\mathbb{Z}[i]$

I am studying number fields and their rings of integer and I want to understand prime ideals of Gaussian integers $\mathbb{Z}[i]$, which is the ring of integers of $\mathbb{Q}[i]$. We have the ...
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Example of fully ramified extension of local field

For $Q_p$, the unramified extension is adding roots of 1 of order prime to p. So what will be the fully ramified extension of $Q_p$? Thanks!
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Extension of $\mathfrak{p}$-adic valuation

I'm trying to understand the relationship between extensions of valuations and extensions of prime ideals. Let $\mathcal{o}$ a Dedekind domain, $K$ its field of fractions, $L|K$ a finite algebraic ...
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1answer
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Factoring primes over rings of integers

Let $m$ be a square free integer and $p$ an odd prime which divides $m$. I wish to factor the ideal $(p)$ over the ring of integers of $\mathbb K = \mathbb Q(\sqrt m)$. I know of an algorithm to use ...
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roots of unity, wild ramification and units of norm one in local fields

Let $\mathbf{K}$ be a finite Galois extension of $\mathbf{Q}_p$ containing the $p$-th roots of unity and let $\mathfrak{p}$ denote the unique prime of $\mathbf{K}$ lying above $p$. For each $n \geq 1$ ...