Questions tagged [ramification]
Ramification in algebraic number theory means prime numbers factoring into some repeated prime ideal factors.
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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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Split, Inert and Ramification points on $\mathbb{P}$
I have the following elliptic curve $E=${$y^2=x^3-x$}$\cup${$\cal{O}$} and $\phi:E\to\mathbb{P}^1_K$ given by $\phi((x,y))=x$ and $\phi(\cal{O})=\infty$.
I need to compute the sets of split, inert and ...
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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 ...
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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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Totally ramified extension with respect to local vs global
Let $F$ be a number field and let $\mathfrak{p}\in \mathsf{Spec} \: \mathcal{O}_F$. We have a non Archimedean valuation $\nu_\mathfrak{p}\colon F\longrightarrow\mathbb{R}_{\geq 0}$, given by $\nu_p(x):...
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When will two fields have same ramification index over $\mathbb{Q}_p$
My purpose is to decide whether two different field extensions $K_1, K_2$ of $\mathbb Q_p$ have same ramification index or not, provided $[K_1:\mathbb Q_p]=[K_2: \mathbb{Q}_p]$ and $K_2=\mathbb{Q}_p(\...
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the sum of the products of ramification degrees and relative degrees
I am reading Algebraic Number Fields by Gerald Janusz and I get confused about the part in the picture below.
Consider two Dedekind domains $R\subset R'$ with quotient fields $K\subset L$. Let $p$ be ...
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product of conjugates with same valuation
Assume $L/K$ is a normal field extension and that $v$ is a valuation on $L$ (not necessarily discrete), with valuation ring $O_v$.
Let $\alpha\in O_v$, $\alpha\not\in K$, and let $\alpha'$ be a ...
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Ramification Locus and Ramification Index
After searching for some examples on the site, I still don't quite get what how one calculates the ramification loci and indices for a morphism.
Here are a few examples that I think may help me ...
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value group of $E＝\Bbb{Q}_p(p^{1/e})$
I want to find what is a value group of $E＝\Bbb{Q}_p(p^{1/e})$($e$ is positive integer, and this is totally ramified extension of degree $p$).
I know value group of $K＝ \Bbb{Q}_p$ is {$p^a$｜$a∈\Bbb{Z}$...
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Valuation of the different for totally ramified extensions
I'm trying the following two exercises from Andrew Sutherland's MIT Number Theory 1 problem sets.
Fix an odd prime $p$.
Let $L/\mathbb Q_p$ be a totally ramified extension of degree $p$ with ...
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value group of maximal tamely ramified subextension of henselian field
In prop 7.11 of Neukirch's Algebraic Number Theory says The maximal tamely ramified subextension $V/K$ of $L/K$ has value group $w(V^*)=w(L^*)^{(p)}$, where $L/K$ is an algebraic extension of ...
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The inertia field is ramified?
I can't seem to figure out how this example is possible. When I put the following into SageMaths:
...
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How does ramification groups help to study ramifications of local field extension?
I am studying higher ramification groups of local field extension.
In Wikipedia it is mentioned that higher ramification groups gives information about ramification of extension.
Suppose $L/K$ be a ...
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Do ramification groups contain non-central abelian normal subgroups?
I am studying the proof of integrality of the conductors of Galois representations from these notes, and I have hit a roadblock in a step of the proof of Proposition 3.1.40 (page 57).
The setting is ...
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Confusion about sum of ramification indices
In algebraic number theory, I'm well aware of the following formula: Given the "$AKLB$ setup" and a prime $\mathfrak{p}$ of $A$, then
\begin{equation}\sum_{\mathfrak{P}|\mathfrak{p}} e_{\...
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How to find the "relative" defining polynomial of an extension of number fields?
I'm looking at the number field $L$ with label 16.0.3243658447265625.1 in the LMFDB, with defining polynomial
$$f= x^{16} - 5 x^{15} + 14 x^{14} - 30 x^{13} + 57 x^{12} - 100 x^{11} + 157 x^{10} - 215 ...
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Decomposition of unramified primes in normal closure of pure quintic field
Let $\Gamma$ = $\mathbb{Q}(\sqrt[5]{n})$ a pure quintic field, $k$ = $\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field, then $N$ = $\mathbb{Q}(\sqrt[5]{n}, \zeta_5)$ is the normal closure of $\Gamma$...
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Maximal Unramified subextension of Compositum
Let $L/K$ and $M/K$ be two finite extensions of henselian fields. Let $T$ and $U$ be the maximal unramified subextensions of $L/K$ and $M/K$, respectively. Is it true that the maximal unramified ...
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Ramification index example.
I was studying ramification index and I found the following definition:
Let $E/K$ a finite field extension, $\mathfrak{d}$ a DVR of $K$ with maximal ideal $m$ and $\mathfrak{D}$ a DVR of $E$ with ...
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Integral basis for totally ramified extensions of $p$-adic fields
Let $L/K$ be a totally ramified extension of $p$-adic fields of degree $n$. Write $v_L$ for the normalised valuation of $L$. If $a_0,\ldots, a_{n-1}$ are elements of $L$ such that $v_L(a_i)\equiv i\...
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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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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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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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