Question

Show that $K$ can be generated over $F$ by the root of an irreducible quartic polynomial of the form $x^4+bx^2+c$.

score 468 · Answer 1 · 2026-03-25 21:51:11

468

Views

Show that $K$ can be generated over $F$ by the root of an irreducible quartic polynomial of the form $x^4+bx^2+c$.

Published on 25 Mar 2026 - 21:51

score 765 · Answer 2 · 2026-03-25 11:53:18

765

Views

Reference request: an introduction to Galois theory (without group/ring theory)

Published on 25 Mar 2026 - 11:53

score 101 · Answer 3 · 2018-05-16 22:40:36

101

Views

Normal Closure of a Field Extension

Published on 16 May 2018 - 22:40

score 52 · Answer 4 · 2018-05-17 01:54:51

52

Views

For a special $\rho \in \mathbb{C}$ $1+ \rho^k+\rho^{2k}+\rho^{3k}+...+\rho^{(n-1)k}=1$ if $n$ divides $k$.

Published on 17 May 2018 - 1:54

score 1.1k · Answer 5 · 2018-05-17 02:10:18

1.1k

Views

Is this Galois Group is isomorphic to $S_5$

Published on 17 May 2018 - 2:10

score 67 · Answer 6 · 2026-03-27 07:18:31

67

Views

Coimage of closed elements and coimage for galois connection between $Quot(X)^{op}$ and $Rel(X)$

Published on 27 Mar 2026 - 7:18

score 342 · Answer 7 · 2026-03-25 21:51:15

342

Views

Let $w=e^{2\pi i/7}$. Find the lattice of subgroups of Gal$(\mathbb Q(w)/\mathbb Q)$ and of intermediate fields in $\mathbb Q(w)\supseteq\mathbb Q$

Published on 25 Mar 2026 - 21:51

score 220 · Answer 8 · 2026-03-25 21:51:11

220

Views

Find the Galois group of $\Bbb Q(\zeta_n,\sqrt[n]{2})/\Bbb Q(\zeta_n)$

Published on 25 Mar 2026 - 21:51

score 727 · Answer 9 · 2026-03-25 21:54:14

727

Views

$K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal (Proof verification)

Published on 25 Mar 2026 - 21:54

score 528 · Answer 10 · 2026-03-25 21:54:05

528

Views

Let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$.

Published on 25 Mar 2026 - 21:54

score 143 · Answer 11 · 2018-05-18 17:07:26

143

Views

finding $[K(α):K] $, and $\operatorname{Aut}(K(α)/K)$ when $K=F_3(t)$ and $α$ is root of the polynomial $f(x)=x^6−tx^3+1$

Published on 18 May 2018 - 17:07

score 117 · Answer 12 · 2026-03-25 11:27:15

117

Views

Find $G (K/F) $ where $K $ is a splitting field of $t^p-t-a \in F [t] $.

Published on 25 Mar 2026 - 11:27

score 34 · Answer 13 · 2018-05-19 08:24:29

34

Views

In a Galois group of $F(a_1,...,a_n)/F $ with $a_i$'s algebraic, where does the $a_i $'s get mapped to

Published on 19 May 2018 - 8:24

score 24 · Answer 14 · 2018-05-19 21:40:57

24

Views

Let $F/K,F'/K'$ be two galois extensions(maybe infinite), $F\subset F',K\subset K'$, is $G(F'/K')\to G(F/K)$ via restriction open map?

Published on 19 May 2018 - 21:40

score 29 · Answer 15 · 2026-03-25 09:50:19

29

Views

A set of generators in disjoint cycle notation. Сompute the Galois group of an irreducible polynomial

Published on 25 Mar 2026 - 9:50

List Question

Show that $K$ can be generated over $F$ by the root of an irreducible quartic polynomial of the form $x^4+bx^2+c$.

Reference request: an introduction to Galois theory (without group/ring theory)

Normal Closure of a Field Extension

For a special $\rho \in \mathbb{C}$ $1+ \rho^k+\rho^{2k}+\rho^{3k}+...+\rho^{(n-1)k}=1$ if $n$ divides $k$.

Is this Galois Group is isomorphic to $S_5$

Coimage of closed elements and coimage for galois connection between $Quot(X)^{op}$ and $Rel(X)$

Let $w=e^{2\pi i/7}$. Find the lattice of subgroups of Gal$(\mathbb Q(w)/\mathbb Q)$ and of intermediate fields in $\mathbb Q(w)\supseteq\mathbb Q$

Find the Galois group of $\Bbb Q(\zeta_n,\sqrt[n]{2})/\Bbb Q(\zeta_n)$

$K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal (Proof verification)

Let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$.

finding $[K(α):K] $, and $\operatorname{Aut}(K(α)/K)$ when $K=F_3(t)$ and $α$ is root of the polynomial $f(x)=x^6−tx^3+1$

Find $G (K/F) $ where $K $ is a splitting field of $t^p-t-a \in F [t] $.

In a Galois group of $F(a_1,...,a_n)/F $ with $a_i$'s algebraic, where does the $a_i $'s get mapped to

Let $F/K,F'/K'$ be two galois extensions(maybe infinite), $F\subset F',K\subset K'$, is $G(F'/K')\to G(F/K)$ via restriction open map?

A set of generators in disjoint cycle notation. Сompute the Galois group of an irreducible polynomial

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