Questions tagged [galois-theory]

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 (galois-connections).

Galois theory is an area of abstract algebra introduced by Evariste Galois, which provides a connection between field theory and group theory. Given a field $F$ and an extension $F$ of $E$ with certain properties (a type of extension called Galois extension), let $\operatorname{Gal}(E/F)$ be the group of automorphisms $\varphi$ of $E$ which leave $F$ fixed, i.e. $(\forall x\in F):\varphi(x)=x$. The fundamental theorem of Galois theory asserts that there is a one-to one correspondence between subfields of $E$ which are extensions of $F$ and subgroups of $\operatorname{Gal}(E/F)$:

  • if $H$ is a subgroup of $\operatorname{Gal}(E/F)$, then the set of those $x\in E$ such that $(\forall\varphi\in H):\varphi(x)=x$ is a subfield of $E$ which is an extension of $F$;
  • to each subfield $K$ of $E$ which is an extension of $F$, one can associate the subgroup of $\operatorname{Gal}(E/F)$ whose elements are those $\varphi\in\operatorname{Gal}(E/F)$ such that $(\forall x\in K):\varphi(x)=x$.
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How to show that $\sqrt{2}+\sqrt{3}$ is algebraic?

In Abbot's Understanding Analysis I am asked to show that $\sqrt{2}+\sqrt{3}$ is an algebraic number. I have shown that those two are algebraic separately (that was simple), but I can't figure out what to do to show that their sum is algebraic, too.…
confused
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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 polynomial in terms of its coefficients assuming the…
user961
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closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic closed-form formulas. And I remember reading somewhere…
user7530
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How to transform a general higher degree five or higher equation to normal form?

This is a continuation of How to solve fifth-degree equations by elliptic functions? How to transform a general higher degree five or higher equation to normal form? For example, a quintic equation to Bring-Jerrard form?
ziang chen
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Galois Group of $x^p - 2$, $p$ an odd prime

Question is to determine the Galois group of $x^p-2$ for an odd prime $p$. For finding the Galois group, we look for the splitting field of $x^p-2$ which can be seen as $\mathbb{Q}(\sqrt[p]{2},\zeta)$ where $\zeta$ is a primitive $p^{th}$ root of…
user87543
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Why does this pattern of "nasty" integrals stop?

We have (typo corrected), $$\begin{aligned} \pi &=\int_{-\infty}^{\infty}\frac{(x-1)^2}{\color{blue}{(2x - 1)}^2 + (x^2 - x)^2}\,dx,\quad\text{(by Mark S.)}\\[1.8mm] \pi &=\int_{-\infty}^{\infty}\frac{(x+1)^2}{\color{blue}{(x + 1)}^2 + (x^2 +…
Tito Piezas III
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Inseparable, irreducible polynomials

The standard examples of irreducible, inseparable polynomials that one encounters in an introductory course on field theory all seem to have only a single root in an algebraic closure. Are there elementary examples of inseparable, irreducible…
user15464
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Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= \sqrt[3]{\tfrac{5-3\sqrt[3]7}2} \end{equation} (see e.g.…
Grigory M
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Why $E\otimes_KE\cong EG$ implies that Galois theory works?

I am reading the book Algebra, volume 1: Fields and Galois theory by Falko Lorenz. This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the Galois group of $E/K$. Let $EG$ be the group algebra…
user45765
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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 expressions in $x$? The easiest example is a pure…
Jack Schmidt
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Galois group of a reducible polynomial over $\mathbb {Q}$

Let $f \in\mathbb {Q}[X]$ be reducible - for the sake of simplicity, $ f = gh$ with $g,h \in\mathbb {Q}[X]$ irreducible. Let L be the splitting field of f. Does $Gal(f) \simeq Gal(g) \times Gal(h)$ hold, were Gal(~) denotes the galois group of the…
Eli
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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 extensions of $\mathbb F_p$, so the Galois group of…
user8268
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Need help understanding separable polynomials

I have the following definition of what it means for a polynomial to be separable: Let $E$ be a field and $P \in E[x]$ be irreducible. Then we say $P$ is separable if it has no repeated root in any field extension of $E$. If $P\in E[x]$ is…
nullUser
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The radical solution of a solvable 17th degree equation

(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients: $$\begin{align*} x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 x^{11}-x^{10}\\ &\quad-4 x^9-11 x^8-7 x^7-13…
Tito Piezas III
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Exercises on Galois Theory

I need a source for exercises on classical Galois Theory, or to be more specific, Galois extensions of finite fields and the rationals as well as applications (solvability by radicals, for example). So far, I have worked with Tignol's "Galois Theory…