# 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).

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### Compute the Galois group for $f(x) = (x^2-p)(x^2-q)(x^2-pq)$ over $\mathbb{Q}$ and determine all subfields of splitting field

For $f(x) = (x^2-p)(x^2-q)(x^2-pq) \in \mathbb{Q}[x]$ where $p\neq q$ are primes I need to compute the Galois group for $f$ over $\mathbb{Q}$ and determine all subfields of the splitting field. Here ...
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### What is the lattice diagram of the Galois extension $\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$?

Consider the Galois extension $K:=\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$. As it is clear, it is of degree $8$ extension. So the Galois group is a group of order $8$. I ...
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### How the Frobenius behaves under the change of base field

Let $k'/k$ be an extension of finite fields, $X$ be a scheme over $k'$ and thus over $k$. Then $X\otimes_{k} \bar{k}$ is $n = [k':k]$ disjoint union of $X\otimes_{k'} \bar{k}$. How the Frobenius of ...
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### Practicing degree arguments for field extensions

Right now I'm just trying to get a better grip on my understanding for prelims. The question is as follows: Suppose we have the polynomial $x^6 -5$, Prove this is irreducible over $\mathbb{Q}$ What ...
Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$. Is there any possibility that two (or more) different $f(z)$ can ...
Find the degrees of $\Bbb{Q(a)/Q, Q(a,b)/Q, Q(a,b,c)/Q}$ where $a,b,c$ are the roots of $x^3+x-1$ where a is real. To solve this problem, I noticed that $x^3+x-1$ is irreducible. Then why isn’t the ...