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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Given only the minimal polynomial of a field and approx primitive element, find subfields

I want to solve the following problem. Given: The minimal polynomial of a number field, and a decimal approximation of the field's primitive element. Compute: Is the number field an imaginary quadratic extension of a real field? More details: $K$ is…
j0equ1nn
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Let $\mathbb{K}$ the the splitting field of $x^4 -2 x^2 -2$ over $\mathbb{Q}$. Determine all the subgroups of the Galois group

Let $\mathbb{K}$ the the splitting field of $x^4-2x^2-2$ over $\mathbb{Q}$. Determine all the subgroups of the Galois group and give their corresponding fixed subfields of $\mathbb{K}$ containing $\mathbb{Q}$. $\mathbb{K}$=…
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Is this 7-th degree polynomial solvable?

Unluckily I don't have a solid background in Galois theory; I ran into this polynomial equation: \begin{equation} x^7-\frac{1}{2}x^6-\frac{3}{2}=0 \end{equation} I know it has only one positive real root, so the other $6$ must be complex conjugates.…
marco trevi
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How to show that the trace maps a Galois extension to the base field

Let $K$ be a finite extension of the finite field $F$, then the trace is defined as $$\operatorname{Tr}(\alpha) = \sum_{\sigma \,\in\, \operatorname{Gal}(K/F)}\sigma(\alpha)$$ How can one show that $\operatorname{Tr}(\alpha) \in F, \forall \alpha…
Anfänger
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Zeroes of irreducible two variables polynomial which is reducible in algebraic closure

Let $\mathbb{F}$ be a field of characteristic $0$, $\bar{\mathbb{F}}$ its algebraic closure, $p(x,y) \in \mathbb{F}[x,y]$ an irreducible polynomial which is reducible in $\bar{\mathbb{F}}[x,y]$. Show that $p(x,y)$ has only a finite number of zeroes…
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How do these explanations for a Galois Group tie in at all?

In relation to my previous question (which I will get back), I am rather very confused with the various explanations people and textbooks have. Firstly, I noticed there are $2$ different Galois Groups(or are they?), $1.$ $Gal_{\mathbb{A}}(f)$ where…
John Trail
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When is $i$ contained in $Q(\zeta)$

As part of a problem I have to determine for which values of $n$ is $i$ contained in $\Bbb Q(\zeta)$, were $\zeta$ is a primitive $n^{th}$ root of the unity. Clearly, if $n$ is multiple of $4$, it is contained since it is in fact a primitive root of…
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How to tell if a set is open in the Krull topology?

I'm an undergraduate not very familiar with topology trying to understand the so called Krull Topology in the context of infinite Galois Theory. We proceed as follows: Let $\Omega/k$ be a (possibly infinite) Galois extension with group…
Shoutre
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$x^4 -10x^2 +1 $ is irreducible over $\mathbb Q$

I have seen the thread Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$ but this didn't really have a full solution. Is it true that if it is reducible then it can be factored into a linear factor or quadratic factor in the form $x^2 - a$.…
snowman
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Is the tensor product of 2 finite extension of $\Bbb Q$ isomorphic to a direct sum of fields?

I have $K_1$ and $K_2$ two finite extensions of $\Bbb Q$. I can construct $K_1 \otimes_\Bbb Q K_2$. This is clearly isomorphic to a direct sum of field as vector space (indeed one can easily see that it is isomorphic to a direct sum of $n_1 \times…
Thalanza
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Inseparable extensions

Given that $F/K$ is a finite extension of fields and the extension is not separable. My question is whether we can always find a subfield $L$ such that $K\subset L \subset F$ and $[F:L]=p$, where $p=char(K)$.
user2902293
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Primitive element of $\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q}$

Is there a clever way to determine a primitive element of the finite extension $$F=\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q} \text{ ?}$$ On simpler examples, I've been able to find one by determining all field morphisms $\sigma: F\to\mathbb{C}$…
Klaus
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General technique for finding minimal polynomial?

I always have a lot of trouble with these problems, "find the minimal polynomial of {number} over {field}" What are the general procedures for solving problems of this format? Thank you for your help
thinker
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Minimal polynomial over $\mathbb Q(\sqrt{-2})$

Find the minimal polynomial for $\sqrt[3]{25} - \sqrt[3]{5} $ over $\mathbb Q$ and $\mathbb Q(\sqrt{-2})$. I have done the first part of this, over $ Q$, and have a polynomial. But I do not know how to do this over $ Q \sqrt{-2}$
thinker
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Counter example of equivalence between separable polynomial and square-free polynomial

Over a perfect field $k$, the notions of separable and square-free polynomial are equivalent. Indeed if $b^2|P$, then $P$ has a repeated root in $\bar k$. Conversely, if $p$ has a repeated root in $\bar k$, then it cannot be irreducible, since $k$…
Rodrigo
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