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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Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\ldots,r_n)/\mathbb{Q})$$ What is the probability, as…
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How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?

This is Exercise 18.14 from Algebra, Isaacs. $p_{1}\ ,\ p_{2}\ ,\ ... p_{n}$ are different prime numbers. How to show that $$\mathbb{Q}[\sqrt{p_{1}}, \sqrt{p_{2}}, \ldots, \sqrt{p_{n}} ] = \mathbb{Q}[\sqrt{p_{1}}+ \sqrt{p_{2}}+\cdots + \sqrt{p_{n}}]…
WLOG
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Why there is much interest in the study of $\operatorname{Gal}\left(\overline{\mathbb Q}/\mathbb Q\right)$?

Let's start for a simple quote from wikipedia: "No direct description is known for the absolute Galois group of the rational numbers. In this case, it follows from Belyi's theorem that the absolute Galois group has a faithful action on the…
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If $\widetilde{L} =$ splitting field of all irreducible polynomials over $L$ of prime-power degree, is $\widetilde{\Bbb{Q}} = \overline{\Bbb{Q}}$?

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 $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? My money is on "no", because I see no obvious reason…
Bruno Joyal
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"Standard" ways of telling if an irreducible quartic polynomial has Galois group C_4?

The following facts are standard: an irreducible quartic polynomial $p(x)$ can only have Galois groups $S_4, A_4, D_4, V_4, C_4$. Over a field of characteristic not equal to $2$, depending on whether or not the discriminant $\Delta$ is a square and…
Qiaochu Yuan
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Is there active research in Galois Theory?

I recently decided to introduce myself to the field of Modern Algebra - in particular, Galois theory - and I found it absolutely beautiful! Thus I would really like to study something in Galois theory, which leads me to ask - do people still develop…
Gauss
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Are most rational quintics unsolvable?

It is well-known that, as polynomials of degree exceeding 4, there exist quintics whose roots cannot be solved for by radicals (Abel-Ruffini theorem). So we can divide the set of rational quintics into those which are solvable by radials and those…
Semiclassical
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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 appears as follows : " The original work of Galois is…
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Why $\sqrt[3]{3}\not\in \mathbb{Q}(\sqrt[3]{2})$?

We all know that $\sqrt[3]{3}\not\in \mathbb{Q}({\sqrt[3]{2}})$ intuitively. We even know that $\sqrt[3]{p}\not\in \mathbb{Q}(\sqrt[3]{q})$ for two distinct primes $p, q$. However, I don't know how to prove these things rigorously. In case of…
Seewoo Lee
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Galois group of $x^3 - 2 $ over $\mathbb Q$

I know the Galois group is $S_3$. And obviously we can swap the imaginary cube roots. I just can't figure out a convincing, "constructive" argument to show that I can swap the "real" cube root with one of the imaginary cube roots. I know that if…
Marty Green
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Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$

So I want to show that $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ and determine its Galois group. My thoughts are as follows: Define $\alpha := \sqrt{2+\sqrt{2}}$. Then it is easily shown that $\alpha$ satisfies…
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Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of "the functions one finds on a calculator"?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. Now $g$ will almost certainly be a function which is…
Kevin Buzzard
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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. I have my own answers and point of view to this…
user1971
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Is it possible for an irreducible polynomial with rational coefficients to have three zeros in an arithmetic progression?

Assume that $p(x)\in \Bbb{Q}[x]$ is irreducible of degree $n\ge3$. Is it possible that $p(x)$ has three distinct zeros $\alpha_1,\alpha_2,\alpha_3$ such that $\alpha_1-\alpha_2=\alpha_2-\alpha_3$? As also observed by Dietrich Burde a cubic won't…
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Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit means we can write down a polynomial equation with…