0
votes
0answers
33 views

Galois representation associated to a number field

I think I'm missing something completely trivial. I want to know how to compute the Galois representation associated to an extension of $p$-adic fields. Let $p$ and $q$ be odd prime numbers. Fix ...
6
votes
1answer
75 views

Is the extension Galois if $\mathrm{Aut}(K)$ acts transitively on the non-ramified prime ideals?

Let $K/\mathbb Q$ be a finite extension such that $\mathrm{Aut}(K)$ acts transitively on the prime ideals that are not ramified above the same prime $p\in\mathbb N$. Is $K$ Galois? Thanks in advance. ...
1
vote
1answer
30 views

maximal abelian extension of exponent $q-1$ of $\mathbb F_q((t))$

I would like to find the maximal abelian extension of exponent $q-1$ of $K=\mathbb F_q((t))$ and find its Galois group. Due to Kummer theory this extension is $K(\sqrt[q-1]{K^*})$ and it's Galois ...
4
votes
1answer
60 views

How to determine the density of the set of completely splitting primes for a finite extension?

In reply of sea turtles comment in this thread Let $k$ be a number field and $K \mid k$ a finite Galois extension. What is the density of the set of completely splitting primes of $k$? As sea turtle ...
1
vote
0answers
38 views

L-function and automorphisms of C

Let $F$ be an element of the Selberg class and $\sigma$ a field automorphism of $\mathbb{C}$ such that $F=\sigma\circ F\circ\sigma^{-1}$. Is the following implication true? ...
8
votes
0answers
126 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
2
votes
0answers
49 views

composition of a $\mathbb{Z}_p$-extensions and Hilbert class field extension (Iwasawa)

I am reading Iwasawa's article On $\Gamma$-extensions of algebraic number fields. In paragraph 7.3 : " We now take as M the maximal unramified abelian $p$-extension of $L$ in $\Omega$ , i.e. the ...
1
vote
2answers
81 views

Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
1
vote
0answers
80 views

Can the error term involved in the PNT be expressed in a Galois theoretic framework?

According to Wikipedia, the current best error term for the prime number theorem is $\pi(x)-\mathrm{Li}(x)=O\left(x\exp\left(-\frac{A(\ln x)^{3/5}}{(\ln\ln x)^{1/5}}\right)\right)$, while RH is ...
1
vote
1answer
47 views

Explicit Kummer isomorphism

Let $K$ be a characteristic $0$ field containing $\mu_n$ (the $n$-th roots of unity). Then it known that the map $K^{\times} / (K^{\times})^n \to \mathrm{Hom}(G, \mu_n)$ which sends $x$ to $\sigma ...
1
vote
0answers
45 views

Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
4
votes
0answers
50 views

Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
1
vote
1answer
86 views

Prime Splits Completely in Every Intermediate Field

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits ...
2
votes
0answers
33 views

About the finiteness of Sha(A/K)

As this question is pretty vague due to my huge lack of knowledge of the subject, it may not be suitable for MathOverflow, and so I prefer to ask it here. If I well understood what I read on ...
0
votes
2answers
76 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
8
votes
2answers
84 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
1
vote
1answer
54 views

A question about cubic roots of rational numbers

I'm trying to understand if, given $K$ a cubic cyclic extension of $\mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a third primitive root of unity, it always exists $b \in \mathbb{Q}$ such that $\sqrt[3]{b} ...
1
vote
0answers
105 views

galois norm and trace of field extensions

Let $K \subset L \subset E$ and let Nm$_{E/K}(x)$ and Tr$_{E/K}(x)$ be its norm and trace, the determinant and trace of $x$ acting by multiplication on $E$. How can one show that $$ ...
0
votes
0answers
47 views

Cyclictomic polynomial is irreducible over $\mathbb{Q}$

http://www.math.uiuc.edu/~r-ash/Algebra/Chapter6.pdf In the site above, section $6.5.5$, in the proof of '$\Psi_n(X)$ is irreducible over $\mathbb{Q}$', first sentence, why the minimal polynomial $f$ ...
4
votes
1answer
105 views

What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
3
votes
1answer
124 views

Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...
5
votes
1answer
57 views

Homomorphisms from the additive groups of virtual characters into certain idele groups

This is a question from Frohlich's book 'Galois Module Structure of Algebraic Integers', Ch.1. Let $K$ be a number field and $\Omega_K=\text{Gal}(K^c/K)$ where $K^c$ is the separable closure of $K$. ...
4
votes
3answers
106 views

Ramification of primes without knowing the discriminant

Let $\mathbb{K} = \mathbb{Q}[\sqrt[3]{5}] \ $, and let $\mathbb{L}$ be the normal closure of $\mathbb{K}$. Let $\mathbb{O}_{\mathbb{K}} \ $ be the integral closure of $\mathbb{Z}$ in $K$ and ...
1
vote
1answer
60 views

Cardinality of prime divisors in cyclotomic fields

1) For $p$ an odd prime, let $K_{n} = \mathbb{Q}[e^{\frac{2\pi i}{p^{n}}}] \ $ , and let $R_{n}$ be the ring of integers of $K_{n}$. Let $q\mathbb{Z} \ $ be a prime ideal of $\mathbb{Z} \ $, with ...
11
votes
1answer
303 views

Galois group over $p$-adic numbers

Can one describe explicitly the Galois group $G=\operatorname{Gal}(\overline{\mathbb Q_p}/\mathbb Q_p)$? I only know the most basic stuff: unramified extensions of $\mathbb Q_p$ are equivalent to ...
1
vote
1answer
519 views

Applying Extended Euclidean Algorithm for Galois Field to Find Multiplicative Inverse

I was trying to apply the Extended Euclidean Algorithm for Galois Field. Among the many resources available, I found the methodology outlined in this document easy to grasp. The above works fine when ...
5
votes
1answer
249 views

Why do no prime ideals ramify in the extension $\mathbb{Q}(\sqrt{p }, \sqrt{q})/\mathbb{Q}(\sqrt{pq })$?

Let $p,q $ be odd integer primes, $p \equiv 1 \pmod 4$ and $q \equiv 3 \pmod 4$. $K = \mathbb{Q }[\sqrt{pq }]$, $L = \mathbb{Q}[\sqrt{p }, \sqrt{q} ]$. Why a prime ideal in $O_{K}$ doesn't ramify in ...
1
vote
1answer
274 views

Patterns in $GF(2)$ Polynomial division.

I am testing Prime polynomials in $GF(2)$ and have noticed a pattern that I hope will help. There's a calculator here if you want to familiarise yourself with polynomials over $GF(2)$. I am testing ...
1
vote
1answer
45 views

Confirm the meaning of Prime and Primitive in a Galois(2) polynomial.

Here it discusses primality (or more accurately irreducibility) and primitivity of ...
1
vote
2answers
55 views

GgT, (polynomial) division and finite fields…

Exercise: Let $f,g \in \mathbb{Z}_2[x]$ be the polynomials $f = x^6 + x^5 + x^4 + 1$ and $g = x^5 + x^4 + x^3 + 1$. Has the diophantic equation $f u + g u = x^4 + 1$ solutions $u,v \in \mathbb{Z}[x]$? ...
1
vote
1answer
68 views

question on Galois theory

Can anybody help me with the following question ? I start with a number field $F/\mathbb{Q}$ which is abelian (that is, a Galois extension of abelian Galois group). I know by the Kronecker-Weber ...
3
votes
1answer
100 views

Why don't I end up with the same splitting field?

I've understood that the splitting field of $x^4+2$ and the splitting field of $x^4-2$ over $\mathbb{Q}$ are both the field $\mathbb{Q}(\sqrt[4]{2} , i)$. With degree $8$ over $\mathbb{Q}$. This ...
5
votes
0answers
47 views

Galois automorphisms and Field extensions [duplicate]

Let $\alpha$ be a root of $x^3+3x-1$ and let $\beta$ be a root of $x^3-x+2$. I want to show that $\alpha^2+\beta$ has degree $9$. There are many ways to do this, but I wish to solve the problem ...
7
votes
1answer
218 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
4
votes
1answer
310 views

Square roots of integers and cyclotomic fields

For every $ N \in \mathbb Z$ there exists an integer $n$ such that $ \sqrt N \in \mathbb Q(\zeta_n)$. I am struggling where to start this question, please suggest me few hints.
1
vote
0answers
97 views

Understanding Andrew Wiles proof of Fermat's last theorem [duplicate]

Possible Duplicate: knowledge needed to understand Fermat’s last theorem proof What are the prerequisites needed to understand wiles proof? Could someone sketch a roadmap of how it could be ...
1
vote
1answer
132 views

Cyclotomy in extensions of $\mathbb{Q}_p$

Let $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Denote by $\zeta_{p^n}$ a primitive $p^n$-th root of unity (where $n$ is a positive integer). Assume that $K$ contains ...
1
vote
0answers
48 views

absolute galois group of Puiseux series with coefficients in $\bar{\mathbb{F}_p}$

Let $K$ be the field of Puiseux series with coefficients in $\bar{\mathbb{F}_p}$ (the algebraic closure of the field with elements). What is the absolute Galois group of $K$ ? Thank you to anyone ...
2
votes
1answer
136 views

Galois group of maximal $p$-extension of $\mathbb{Q}_l(\zeta_p)$?

Let $F$ be the field obtained by adjoining to $\mathbb{Q}_l$ a $p$-th root of unity, with $p \not = l$. Denote by $F(p)$ the maximal $p$-extension of $F$, i.e. the maximal extension $L:F$ such that ...
2
votes
1answer
54 views

How many prime ideals are fixed by a given permutation?

Suppose $L$ is a finite Galois field extension of the rational number field $\mathbb{Q}$, and $B$ is the integral closure of $\mathbb{Z}$ in $L$. Let $\sigma$ be an element of the Galois group ...
2
votes
1answer
103 views

Minimal polynomial of the form $\zeta_p+\frac{1}{\zeta_p}+\zeta_q+\frac{1}{\zeta_q }$?

We can calculate the minimal polynomial of $ 2cos(\frac{2\pi}{7})=\zeta_7+\frac{1}{\zeta_7}$ over Q as x^3+x^-2x-1 and simlary for $2cos(\frac{2\pi}{5})=\zeta_5+\frac{1}{\zeta_5 }$. Now my question ...
0
votes
1answer
51 views

What is this quadratic form as a invariant of Galois Extensions?

Suppose that $E/F$ is a Galois extension and viewing E as a vector space over $F$, then quadratic from $Tr_F^{E}(\alpha^2)(\alpha\in E)$ carries some information of the extension. My question is that, ...
4
votes
2answers
306 views

Proving the Möbius formula for cyclotomic polynomials

We want to prove that $$ \Phi_n(x) = \prod_{d|n} \left( x^{\frac{n}{d}} - 1 \right)^{\mu(d)} $$ where $\Phi_n(x)$ in the n-th cyclotomic polynomial and $\mu(d)$ is the Möbius function defined on the ...
3
votes
2answers
398 views

Linear independence of roots over Q

Let $p_1,\ldots,p_k$ be $k$ distinct primes (in $\mathbb{N}$) and $n>1$. Is it true that $[\mathbb{Q}(\sqrt[n]{p_1},\ldots,\sqrt[n]{p_k}):\mathbb{Q}]=n^k$? (all the roots are in $\mathbb{R}^+$) ...
2
votes
2answers
258 views

Are absolute Galois groups compact topological groups

Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$. We have three Galois groups: $G_{\mathbf{Q}} = ...
1
vote
1answer
89 views

Finding generators of cubic Kummer extensions

Let $K$ be a number field containing $\mu_3$, the third roots of unity. Consider a monic irreducible cubic polynomial $f \in K[x]$ whose discriminant $\Delta$ is a square in $K$. Thus the splitting ...
5
votes
1answer
413 views

Realizing $S_n$ as a Galois group

My question is about the realization of the symmetric group $S_n$ as a galois group of a real and normal field extension $K/\mathbb Q$. As I read, such a field $K$ can be obtained as the splitting ...
10
votes
1answer
200 views

Is $\operatorname{Gal}(\mathbb{Q}_p^{un})\cong \hat{\mathbb{Z}}$?

Is the absolute Galois group of $\mathbb{Q}_p^{un}$ the profinite completion of $\mathbb{Z}$? I was never quite sure... In similar cases, it is true. Namely, $\mathbb{C}((t))$ does have absolute ...
20
votes
2answers
1k views

Original works of great mathematician Évariste Galois

Through this question I wanted to know the original works of Galois. When I was reading Galois theory ( since from last month ) , I have been seeing one common line in every book, whose essence ...
1
vote
0answers
120 views

Basic example of extensions of residue fields.

Can anyone think of a simple example of the following: $B/A$ is an integral extension of DVRs with quotient fields $L$ and $K$ and residue fields $\bar{L}$ and $\bar{K}$, $L/K$ is finite dimensional ...