# Tagged Questions

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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$... 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 ... 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 ... 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 ... 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. ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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]? ... 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 ... 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 ... 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 ... 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 ... 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. 0answers 97 views ### Understanding Andrew Wiles proof of Fermat's last theorem [duplicate] Possible Duplicate: knowledge needed to understand Fermatâ€™s last theorem proof What are the prerequisites needed to understand wiles proof? Could someone sketch a roadmap of how it could be ... 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 ... 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 ... 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 ... 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 ... 1answer 104 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 ... 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, ... 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 MÃ¶bius function defined on the ...
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 $\mathbb{R}^+$) ...