# 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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### Checking if the intersection of two cyclic $p$-adic extensions with certain properties is trivial

Let $L$ and $L'$ be finite extensions of $K = \mathbb{Q}_p$. Also, let $n = [L:K]$ and $e = e(L/K)$. Furthermore, we assume the following properties: $L$ and $L'$ are both cyclic over $K$, $L'/K$ is ...
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### Infinite sums for quintic polynomial

It is well known that there is no finite solution for the roots of a quintic polynomial. Are there any nice formulas in terms of infinite sums? Clearly the definition of nice is important for the ...
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### How does Discriminant of splitting field of irreducible polynomial $f$ related to discriminant of $f$ ??

Can we express Discriminant of splitting field of polynomial $f$ in terms of the discriminant of $f .$
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### Finding a fixed polynomial under the multiplicative inversion automorphism

Can anyone find a polynomial $f ∈ ℚ\left(X+\frac{1}{1-X} + \frac{X-1}{X}\right) ⊆ ℚ(X)$ that is fixed under the automorphism $(X ↦ \frac{1}{X})$? $f = X+\frac{1}{X}$ would be nice, but I don't know ...
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### Show that $H_1$ and $H_2$ are conjugate subgroups of $G=\text{Gal}(L/K)$.

Suppose that $L$ is a finite Galois extension of $K$, $f$ is a monic irreducible polynomial in $K[X]$, and $\alpha_1$ and $\alpha_2$ are elements of $L$ such that $f(\alpha_1)=f(\alpha_2)=0$. Prove ...
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### What finite groups can appear as collections of automorphisms of some field? Proof verification

What finite groups G can appear as collections of automorphisms of some field? More precisely, for which G does there exist a field F such that G is a subgroup of the automorphism group of F? What ...
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### An extension corresponding to a subgroup of Galois group

Let $G$ be the Galois group of $f(x)=x^6-2x^4+2x^2-2$ over $\mathbb{Q}$. Describe an extension corresponding to any of it's proper subgroups of maximal order (i.e. find generators of this extension). ...
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### How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$
This is somewhat of a follow-up to this question: A complete picture of the lattice of subfields for a cyclotomic extension over $\mathbb{Q}$. After reading this, I am still confused on how to find ...