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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Higher ramification groups of Galois extension of order $p^2$

Let $p\in \mathbb{Z}$ be a prime number and $K/\mathbb{Q}$ be a Galois extension of degree $p^2$ over $\mathbb{Q}$. Suppose that $P\subset \mathcal{O}_K$ is the only prime ramified over $p$. Let $G:=\text{Gal}(K/\mathbb{Q})$ and recall the…
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Dihedral groups are solvable

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get a solvable group.
Joey
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$f$ is irreducible $\iff$ $G$ act transitively on the roots

Let $F$ be a field and $f\in F[X]$ a separable polynomial. Let $K_f$ be the splitting field of $f$ and $G_f=\text{Gal}(K_f/F)$ its Galois group. Show that $f$ is irreducible $\iff$ $G_f$ acts transitively on the roots on $f$. My work $\implies :$…
MSE
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$p$-Sylow subgroups of a group of order $5^3\cdot 29^2$

I need to calculate the $p$-Sylow subgroups of a Galois group with order $5^3 \cdot 29^2$, i.e. $|\mathrm{Gal}(K/F)|=5^3 \cdot 29^2$. I've already established that there is only one 29-Sylow-subgroup (with $|G_{29}|=29^2$) by the following…
Zachi Evenor
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Algebraic solutions of a differential equation.

Given a differential equation $y' = (1/x)(y^2 + y^3)$. My question is how does one go about finding the solutions of this differential equation which are algebraic over the field $\Bbb{C}(x)$,if any. Notation- $\Bbb{C}(x)$ is the quotient field…
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Formal derivatives over finite fields.

I am slightly confused about what formal derivatives over finite fields mean. Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$. By checking each element of $\mathbb{F}_7$ we easily see that this is irreducible. What about separable? Can we look…
user32134
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Minimal polynomial of $\theta^2$

I'm doing this Galois Theory question: $f = x^3 + x + 3$ is known to be irreducible and has just one real root, call it $\theta$. What is the minimal polynomial for $\theta^2$? Is it just $f$?
John.P
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Is an unramified cover of the p-adics determined by its degree?

If $K_1$ and $K_2$ are subfields of a pre-chosen $\overline{\mathbb{Q}_p}$, and if they're both unramified at $p$, and $[K_1:\mathbb{Q}_p]=[K_2:\mathbb{Q}_p]$, does that imply that $K_1=K_2$? My intuition says that this is true because all that's…
Nicole
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Can I square the triangle?

I know I can't construct a square with the same area as a given circle (because $\pi$ is transcendental). Can I construct (ruler and compass) a square with the same area as a given triangle? I think I can because Heron's formula only includes a…
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For $K$ the splitting field of $x^8+1$ over $\mathbb{Q}$, determine $Gal(K/\mathbb{Q})$.

Let $f(x) = x^8+1$. To determine the Galois group $G$, we first need the splitting field and before that we need to find the zeroes of $f$. So, $\left(re^{i\theta}\right)^8 = 0$ implies $r=1, \theta=\frac{\pi}{8}, \frac{3\pi}{8},\ldots,…
Derek Allums
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Equal Field Extensions

Let $K$ be a field, $a_1, a_2, ..., a_n$ be algebraic over $K$, and $c_1, ..., c_n \in K$. My question is regarding the extension $K[a_1, ..., a_n]$. Is it possible to have an extension $K[c_1a_1+c_2a_2+...+c_na_n]$ which is not equal to $K[a_1,…
AnotherPerson
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What Is the Basis of the Splitting Field of $x^3 - 2$ over $\mathbb Q$?

Let $\zeta$ denote the primitive $3$rd root of unity, and the roots of the polynomial are $ \sqrt[3]2, \zeta \sqrt[3]2, \zeta^2 \sqrt[3]2$. From this, I cannot see how the splitting field has degree $6$ over $\mathbb Q$. However, I can prove that…
Andy Tam
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How to describe when a simple extension $F(\alpha)/F$ is Galois in terms of the minimal polynomial of $\alpha$?

I have a question concerning definition in terms of minimal polynomial i.e. if we let $E = F(\alpha)$ be a field extension of $F$ of degree two then how do I describe, in terms of the minimal polynomial for $\alpha$ over $F$ when this field…
Karl
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Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reading texts about field extensions I've come across the following two definitions for the simple extension $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ is the smallest field containing both $K$ and…
Edward Hughes
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showing that two simple field extensions of $\mathbb{Q}$ are not isomorphic

How can I show that $\mathbb{Q}[x]/(x^2+2)$ and $\mathbb{Q}[x]/(x^2-2)$ are not isomorphic? If $\alpha$ and $\beta$ are zeros of $x^2+2$ and $x^2-2$ in certain extensions respectively, we have that $\mathbb{Q}[x]/(x^2+2)\cong\mathbb{Q}(\alpha)$…
Chilote
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