Tagged Questions

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-...

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If $K$ is a Galois extension of $Q$, can we find $E$ so that $Gal(E/K)$ = $S_n$, for all $n$? [duplicate]

If $K$ is a Galois extension of $Q$, can we find $E$ so that $Gal(E/K)$ = $S_n$, for all $n$? edit: I would appreciate if this wasn't closed, the question that was refered to does not provide an ...
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Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Let $f$ be irreducible over $\mathbb{Q}$ with splitting field $F$. Suppose $\text{Gal}(F/\mathbb{Q})$ is either $D_8$ or $Q_8$. What are the possibilities for $\deg f$? I'm using Dummit & Foote,...
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Question about the Fundamental Theorem of Galois Theory

This is something more like a small doubt than some problem that I need help with. I'm doing and exercise that is asking me to find subextensions of a given extension of all posible orders, and find ...
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Why are these two calculations in $GF(2^5)$ not equal [closed]

I have a quick question about Galois fields, since there seems to be something I have misunderstood way back in university. Addition and subtraction in Galois fields are both done using XOR ...
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Field Extension for which Galois correspondence fails [closed]

Find a non-Galois field extension such that the Galois correspondence fails. Can't seem to come up with a nice answer to this.
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the splitting field for $x^p - 1$

I need to show that the splitting field for a polynomial of the form $f(x) = x^p - 1$, where $p$ is prime and the coefficients of $f$ are in $\mathbb{Q}$, can be expressed as $\mathbb{Q}(\gamma)$, ...
Why doesn't the fundamental theorem of Galois theory apply to the extension $\mathbb{Q}(\sqrt[5]{3})$?
I understand that this field is not a splitting field for any polynomial, as it does not contain the roots of unity. If we had something like $\mathbb{Q}(\sqrt[5]{3}, \gamma)$, where $\gamma$ is a ...
Why if $E$ has characteristic $p$ then $F=\{a\in E:a^{p^n}=a\}$ is a subfield?
We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all ...