List questions from extension-field

Published on 11 Apr 2026 - 15:08

490

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Galois group of $\mathbb{Q}(\sqrt[3]{5}, e^{2\pi i/3})$ over $\mathbb{Q}$ and $\mathbb{Q}(e^{2\pi i/3})$

Published on 17 Apr 2026 - 12:33

562

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Zariski's Lemma

Published on 14 Apr 2026 - 13:10

501

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Let $K \subseteq L \subseteq M$ be two field extensions then, $K \subseteq M$ is algebraic iff $K\subseteq L$ and $L\subseteq M$ are both algebraic.

Published on 16 Apr 2026 - 0:56

148

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Algebra which becomes a field after a base change

Published on 11 Apr 2026 - 7:23

95

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Prove that $\sqrt[7]{2}\not\in\mathbb{Q}(\sqrt[7]{3})$

Published on 14 Apr 2026 - 19:42

911

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Prove that $\sqrt{3} + \sqrt{7}$ is a primitive element of $\mathbb{Q}(\sqrt{3},\sqrt{7})$

Published on 17 Apr 2026 - 8:06

100

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system of intermediate fields of extension is infinite

Published on 12 Apr 2026 - 8:16

38

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Prove that $\operatorname{Gal}(K_1 / F_1)$ and $\operatorname{Gal}(K_2 / F_2) $ are isomorphic

Published on 14 Apr 2026 - 0:32

45

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$E=K(a_1,...,a_n)$ and $\sigma \in Gal(E/K)$ such that $\sigma (a_i)=a_i$ for all $i$. Prove $\sigma = 1_E$

Published on 16 Apr 2026 - 0:04

792

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If $L/K$ is a finite field extension and $L$ is perfect, then $K$ is perfect.

Published on 14 Apr 2026 - 14:26

52

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Finding ordered pair of polynomial $(f(x),g(x))$ with coefficient $\mathbb{Z}_{p}$ for a prime number $p$

Published on 14 Apr 2026 - 2:35

67

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Is the extension $ℚ(\alpha):ℚ(\alpha^2)$ a Galois extension?

Published on 15 Apr 2026 - 3:21

70

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Is $(x-t)^p$ irreducible?

Published on 13 Apr 2026 - 13:33

34

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Galois group is dihedral iff $\exists \beta_i\in F$

Published on 15 Apr 2026 - 17:25

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Cubic radical extension

Galois group of $\mathbb{Q}(\sqrt[3]{5}, e^{2\pi i/3})$ over $\mathbb{Q}$ and $\mathbb{Q}(e^{2\pi i/3})$

Zariski's Lemma

Let $K \subseteq L \subseteq M$ be two field extensions then, $K \subseteq M$ is algebraic iff $K\subseteq L$ and $L\subseteq M$ are both algebraic.

Algebra which becomes a field after a base change

Prove that $\sqrt[7]{2}\not\in\mathbb{Q}(\sqrt[7]{3})$

Prove that $\sqrt{3} + \sqrt{7}$ is a primitive element of $\mathbb{Q}(\sqrt{3},\sqrt{7})$

system of intermediate fields of extension is infinite

Prove that $\operatorname{Gal}(K_1 / F_1)$ and $\operatorname{Gal}(K_2 / F_2) $ are isomorphic

$E=K(a_1,...,a_n)$ and $\sigma \in Gal(E/K)$ such that $\sigma (a_i)=a_i$ for all $i$. Prove $\sigma = 1_E$

If $L/K$ is a finite field extension and $L$ is perfect, then $K$ is perfect.

Finding ordered pair of polynomial $(f(x),g(x))$ with coefficient $\mathbb{Z}_{p}$ for a prime number $p$

Is the extension $ℚ(\alpha):ℚ(\alpha^2)$ a Galois extension?

Is $(x-t)^p$ irreducible?

Galois group is dihedral iff $\exists \beta_i\in F$

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