Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Transcendental Galois Theory

Is there a good reference on transcendental Galois Theory? More precisely, if $K/k$ admits a separating transcendence basis (or maybe if it is a separably generated extension) it seems to me that ...
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596 views

Trace as Bilinear form on a field extension

Can anyone help with this: If $L/K$ is a finite field extension, and we have a $K$-bilinear form given by $$(x,y)\mapsto Tr_{L/K}(xy)$$ then the form is either non-degenerate or $Tr_{L/K}(x)=0$ for ...
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0answers
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Rational independence

The specific question is the following: I am given a set of $[L/2]$ numbers $$g(n) = \sqrt{ c(n)^2 + \alpha c(n) + \beta},$$ where $c(n) = \cos(2\pi n/L)$ (so both $c(n)$ and $g(n)$ depend on $L$ ...
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1answer
531 views

Constructive Proof of Kronecker-Weber?

This question is motivated by my attempt at solving Proving $2 ( \cos \frac{4\pi}{19} + \cos \frac{6\pi}{19}+\cos \frac{10\pi}{19} )$ is a root of$ \sqrt{ 4+ \sqrt{ 4 + \sqrt{ 4-x}}}=x$ Consider ...
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Showing that an intermediate field is not closed

Hungerford defines a field, $E$ as being closed if $E=E''$ where $E'= \{ \sigma \in \mathrm{Aut}(F/K)|\sigma(u)=u \text{ for all } u\in E \} = \mathrm{Aut}(F/E)$ is a subgroup of ...
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Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...
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Explicit computation of a Galois group

Let $E$ be the splitting field of $x^6-2$ over $\mathbb{Q}$. Show that $Gal(E/\mathbb{Q})\cong D_6$, the dihedral group of the regular hexagon. I've shown that $E=\mathbb{Q}(\zeta_6, ...
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2answers
375 views

Solvable subgroups of $S_p$ of order divisible by $p$

This question is from Dummit and Foote's Abstract Algebra, page 638, question 20. It gives a nice paragraph of hints that basically guides one through the problem, but I'm very stuck at a crucial ...
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0answers
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Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
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Galois groups of polynomials and explicit equations for the roots

Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
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The algebraic closure of a finite field and its Galois group

$F$ is an extension field of a field $K$. Let $F$ be an algebraic closure of $\mathbb{Z}_p $ ($p$ prime). Show that $(i)$ $F$ is algebraic Galois over $\mathbb{Z}_p$ $(ii)$ The map ...
3
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1answer
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Where can I find the paper by Shafarevich on the result of the realization of solvable groups as Galois groups over $\mathbb{Q}$?

I ha come across a book on groups as Galois groups, and in the introduction it mentions the paper by I.R. Shafarevich which says that every solvable group can be realized as Galois groups of some ...
6
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1answer
453 views

Splitting Fields

This is a problem from Hungerford's book: Let $E$ be an intermediate field of extension $K\subset F$ and assume that $E=K(u_1, \cdots ,u_r)$ where $u_i$ are (some of the) roots of $f\in K[x]$. Then ...
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Splitting field of $x^{n}-1$ over $\mathbb{Q}$

From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8 Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ ...
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2answers
283 views

What is the condition for a field to make the degree of its algebraic closure over it infinite?

As we all know, the algebraic closure often has an infinite degree. Also, this shows the necessary and sufficient condition for a Galois extension to be a finite extension of fields. However, we may ...
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Relationship between Cyclotomic and Quadratic fields

Since $\varphi(p)=p-1$ is even the p'th cyclotomic field contains some quadratic field. Hecke says that in fact every quadratic field is contained by some cyclotomic field. What is this theorem ...
7
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765 views

A question regarding normal field extensions and Galois groups

The following is possibly true but I can't find a corresponding theorem: If $E/F$ is the splitting field of some polynomial in $F$ and $F \subset K \subset E$ then: $Gal(E/K)$ normal subgroup of ...
12
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3answers
616 views

A question regarding the definition of Galois group

In my book, Galois group is defined to mean the set of automorphisms on $E/F$ that "leave alone" the elements in $F$. On Wikipedia it says: "If $E/F$ is a Galois extension, then $Aut(E/F)$ is called ...
6
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1answer
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Let $F$ be a Galois extension over $\mathbb{Q}$ with $[F:\mathbb{Q}]=2^n$, then all elements in $F$ are constructible

Let $F\subseteq\mathbb{C}$ be a Galois extension of $\mathbb{Q}$ such that $[F:\mathbb{Q}]=2^n$; then all elements in $F$ are constructible. Added. Here is what I have so far. Since there ...
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3answers
571 views

Conjoining elements to fields which are the roots of irreducible polynomials

I know that if $F$ is a field, and $x,y$ are roots of an irreducible polynomial over $F$ lying in an algebraic extension of $F$, then $F(x) \cong F(y)$. But it seems it is not true (in general) that ...
6
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1answer
138 views

Cyclic Extensions of $\mathbb{R}(t)$

Let $\mathbb{R}(t)$ be the field of rational functions over $\mathbb{R}$ (the fraction field of $\mathbb{R}[x]$). I am looking for elements in the Brauer group of the field, and the current idea I ...
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Contributions of Galois Theory to Mathematics

What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories. ...
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Invariant under transformation $i\mapsto -i$ implies real?

When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for ...
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Expressing a root of a polynomial as a rational function of another root

Is there an easy way to tell how many roots $f(x)$ has in $\Bbb{Q}[x]/(f)$ given the coefficients of the polynomial $f$ in $\Bbb{Q}[x]$? Is there an easy way to find the roots as rational ...
4
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1answer
108 views

Is an unramified cover of the p-adics determined by its degree?

If $K_1$ and $K_2$ are subfields of a pre-chosen $\overline{\mathbb{Q}_p}$, and if they're both unramified at $p$, and $[K_1:\mathbb{Q}_p]=[K_2:\mathbb{Q}_p]$, does that imply that $K_1=K_2$? My ...
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1answer
160 views

Field extension question

A question I recall from Carl Linderholm's "Mathematics Made Difficult", Chapter 3 Exercise 8. ...
5
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2answers
231 views

Why are cubics separable over fields that are not characteristic 2 or 3

Why is it that cubics are separable over fields that are not of characteristic $2$ or $3$? This is the starting point for some a discussion of the Galois group of a cubic, but I seem to be stuck ...
10
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1answer
455 views

A different proof of the insolubility of the quintic?

I'm familiar with the "standard" proof using Galois theory that there is no general formula for solving an equation of fifth (or higher) degree using radicals (i.e. arithmetic and root-taking). ...
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1answer
338 views

Galois theory with non-irreducible polynomials

I was browsing through Wikipedia for some reason or another, and it says that there is a way to do Galois theory without worrying about whether or not polynomials are irreducible. That's a bit ...
3
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1answer
113 views

Congruences mod primes in Galois extensions

I have the following situation: let $m,n$ be integers such that $m|n$ and let $\zeta_m$, $\zeta_n$ denote primitive $m$ and $n$th roots of unity. Then we have the inclusion of fields ...
2
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2answers
192 views

Powers of $x$ as members of Galois Field and their representation as remainders

first question on math.stackexchange :) I'm studying for a Cryptography - Communication Security exam, and it involves a certain quantity of number theory - finite field theory, so be warned: this is ...
10
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4answers
479 views

How to describe the Galois group of the compositum of all quadratic extensions of Q?

This is Problem 1.7 from Gouvea's lecture notes on deformations of Galois representations. In particular, he asks you to show that it has many subgroups of finite index which are not closed. So here's ...