# Tagged Questions

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### Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
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### Are there relations among Frobenii?

Let $G=\text{Gal}(\overline{\mathbf Q}/\mathbf Q)$, and for each prime $p$, choose an embedding $\overline{\mathbf Q} \hookrightarrow \overline{\mathbf Q_p}$. Let $\sigma_p$ be a choice of Frobenius ...
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### When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that ...
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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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### Subgroup of class group

Let $K/\mathbb Q$ be a finite Galois extension with Galois group $G$, ring of integers $\mathcal O_K$ and $\mathcal Cl(K)$ its ideals class group. I want to show that $\mathcal Cl(K)^G$ is generated ...
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### Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree. Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
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### Irreducibility of polynomials of the form $x^p - n$ over the cyclotomic field $Q(\zeta_p)$?

Is there a general procedure for showing that the polynomial $x^p - n$ is irreducible over the cyclotomic field $Q(\zeta_p)$? ($\zeta_p$ a primitive pth root of unity, and $n \in \mathbb{N}$. Maybe ...
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### Relating Galois groups related via completions

Let $K$ be a number field, and let $K_\mathfrak p$ denote the completion of $K$ at the prime $\mathfrak p$ of $K$. I'm wondering what can be said (if anything useful) that relates the Galois groups ...
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### Integral closures and Galois extensions

I was reading in Lang's Algebraic Number Theory (Second Edition page 15-16) and the following proposition occured. Proposition 14. Let $A$ be integrally closed in its quotient field $K$, and let $B$ ...
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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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### 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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### Importance of continuity of Galois representations

So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is ...
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### Elements of order 2 in the absolute Galois group

So I remember reading once that the only element of $G=Gal(\overline{\Bbb Q} / \Bbb Q)$ that we understand is complex conjugation. Suppose we fix an embedding of $\overline{\Bbb Q}$ into $\Bbb C$. ...
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### 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 ...
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### 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 ...
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### Parametric quadratic equation in Q(a)

Are there any algebraic numbers $a$ with the following property: for all $q\in\mathbb{Q}$, the square roots of $a+q$ are in $\mathbb{Q}(a)$ ? I`m fairly certain that the answer is no, but how does ...
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### Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
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### When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let ...
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### 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 ...
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### Ramification of a Galois extension

I understand that an extension of number field $L/K$ is unramified if every non-zero prime ideal of $\mathcal{O}_K$ is unramified in $L$ (where a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ is ...
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### 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 ...
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### What is meant by 'the completion of Z'?

In the first chapter of Algebraic Number Theory (lecture notes collected by Cassels-Fröhlich), page 28 has the following paragraph: "We suppose now that $k$ is a finite field of characteristic $p$ ...
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### Symmetric function theorem and Galois Theory — How deep is the connection?

By symmetric function theorem in the title, the fundamental theorem of symmetric polynomials is meant: Any symmetric polynomial has a unique representation as a polynomial in the elementary symmetric ...
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### 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 $K$. ...
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### Trinomial extensions

Let F be the smallest field containing the rationals, such that whenever a and b are in F, and d>e are positive integers, the polynomial $x^d + ax^e + b$ splits in f. How do you prove that F is not ...
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### 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 ...
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### 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 ...
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### How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
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### Number fields containing the values of a character

This is a basic question- Let $K$ be an algebraic number field, $\Gamma$ a finite group, and $R(\Gamma)$ the ring of virtual characters of $\Gamma$ with values in the algebraic closure $\mathbb{Q}^c$ ...
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### Quadratic equation family with largest real root in Cyclotomic extension

Let the $\alpha_{k}$ be the largest real root by absolute value of $2x^2-2kx-(k-1)=0$ for all $k\ge1$. For what values of $k$ does $\alpha_{k}$ sit in a cyclotomic extension? How does one explicitly ...
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### Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
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### Degrees of the real and imaginary parts of an algebraic number

I am working on a theory of generalized geometric constructions, which involves generating new numbers as real roots of polynomials whose coefficients are existing numbers satisfying certain ...
Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...