Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [ramification]

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

Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors. For instance, if $p$ is a rational prime and $K / \Bbb Q$ a number field, then $p$ ramifies in $K$ if $p \mathcal{O}_K$ decomposes as a product of prime ideals $\prod\limits_{i=1}^m P_i^{e_i}$ so that there is some index $i$ such that $e_i \geq 2$.

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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 extension of degree $n$, and let $B$ be the integral…
Pedro
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Understanding the Inertia Group in Ramification Theory

I am a beginner student of Algebraic Number Theory and I am starting to learn ramification theory (of global fields). My question asks for motivation for a definition I was given. Let $K$ be an algebraic number field, $\mathcal{O}_{K}$ its ring of…
Shoutre
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Splitting of a prime in compositum

Let $K$ be a number field. Let $L_1$ and $L_2$ be two extensions of $K$. Let $P\subseteq O_K$ be a prime ideal. We know that if $P$ is unramified in $L_1$ and $L_2$ it remains unramified in the compositum $L_1L_2$. Do we have a similar theorem when…
learning_math
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$\mathbb{Q}(\sqrt[3]{17})$ has class number $1$

Let $\alpha:=\sqrt[3]{17}$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$ I have to show that $K$ has class number $1$, i.e. $\mathcal{O}_K$ is a PID. The…
user72870
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First derivative of the Weierstrass $\wp$ function as a function on $\mathbb{C}/\Lambda$

I am currently trying to prove various facts about $\wp'$, considered as a meromorphic map from $\mathbb{C}/\Lambda\to\mathbb{C}$, where $$\wp'(z) = -2\sum_{w\in\Lambda}\frac{1}{(z-w)^3}.$$ In particular, I am interested in the degree of this map…
Tim
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Computing a uniformizer in a totally ramified extension of $\mathbb{Q}_p$.

Do you know how to compute a uniformizer of $\mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})$? Where $\zeta_{p^n}$ is a primitive $p^n$-th root of 1 and $p$ is an odd prime.
Bear
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A prime number $p$ is ramified in $\mathbb{Q}(\sqrt[p]{a})$.

Let $p$ be an odd prime number and $a\in \mathbb {Z}$ with $\sqrt[p]{a}\notin \mathbb{Z} $. Prove that $p$ is ramified in the number field $\mathbb{Q}(\sqrt[p]{a})$. My idea is to apply Dedekind's Theorem: A rational prime number is ramified in a…
user341609
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Intuition in studying splitting and ramification of prime ideals

I am trying to learn Algebraic Number Theory alone and I'm having serious trouble understanding the ramification and splitting of primes ideals in Galois extensions of a number field $L/K$. Some of my friends tell me lots of facts about these and…
Shoutre
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Decomposition group and inertia group

Let $L/K$ be a Galois extension with Galois group $G$. Let $O_K$ and $O_L$ be the ring of algebraic integers of $K$ and $L$ respectively. Let $P\subseteq O_K$ be a prime. Let $Q\subseteq O_L$ be a prime lying over $P$ with ramification index…
learning_math
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References to understand $K3$ surface as a double cover of $\mathbb{P}^2$ ramified along a sextic

My goal is to understand that $2:1$ cover of $\mathbb{P}^2(\mathbb{C})$ ramified along a sextic is a $K3$ surface. My main problem is in understanding the theory of ramified covering of $\mathbb{P}^2.$ Since $\mathbb{P}^2$ is compact, there are…
user166467
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Ramified prime in cyclotomic extension of a number field

Let $K$ be a number field, $n$ be a positive integer and $\zeta_n$ a primitve $n^{th}$ root of unity. How does one show that if a prime ideal $\mathfrak{p}$ of $K$ is ramified in $K(\zeta_n)$ then $n \in \mathfrak{p}$ ? (This is not homework). In…
Zorba le Grec
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Ramification index and inertia degree same for all the primes, then is the extension Galois

Let $L/K$ be a Galois extension of number fields. We know that if $Q,Q'$ are two primes of $L$ lying over a prime $P$ of $K$, then e$(Q|P)$=e$(Q'|P)$ (the ramification indices are same ) f$(Q|P)$=f$(Q'|P)$ (the inertia degrees are same ) My…
learning_math
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Two definitions of ramification groups, why are they equivalent?

Let $L|K$ be a finite galois extension and suppose that $v_k$ is a discrete normalized (non-archimedean) valuation of $K$ with positive residue field characteristic $p$, and that $v_K$ admits a unique extension $w$ to $L.$ Further, let…
MaxAldrin
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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 $\mathfrak{p}=\mathfrak{b}_{1}^{e}...\mathbb{b}_{g}^{e}$ where $ \mathfrak{b}_{i} \in \mathcal{O}_{L}$…
Aster Phoenix
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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 that it suffices to show that there are no…
Arbutus
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