Question

$|\mathrm{Gal}(L/K)| = 105$ and subgroups $A,B \leq \mathrm{Gal}(L/K)$ with $|A| = 21$ and $|B| = 35$. Prove that $L^{A}\cap L^{B} = K$.

score 75 · Answer 1 · 2018-06-24 16:20:40

75

Views

$|\mathrm{Gal}(L/K)| = 105$ and subgroups $A,B \leq \mathrm{Gal}(L/K)$ with $|A| = 21$ and $|B| = 35$. Prove that $L^{A}\cap L^{B} = K$.

Published on 24 Jun 2018 - 16:20

score 1.2k · Answer 2 · 2018-06-25 20:06:50

1.2k

Views

Constructing a field in which polynomial has root

Published on 25 Jun 2018 - 20:06

score 190 · Answer 3 · 2018-06-26 15:49:50

190

Views

Show that every root of a polynomial is real (using Galois theory)

Published on 26 Jun 2018 - 15:49

score 445 · Answer 4 · 2018-06-26 17:53:08

445

Views

Determine $Gal(K/F)$ and find all the intermediate subfields of $K/F$.

Published on 26 Jun 2018 - 17:53

score 1.2k · Answer 5 · 2018-06-26 18:32:14

1.2k

Views

Let $K$ be a Galois extension of $F$ with $[K/F] = n$. If $p$ is a prime divisor of $n$, show that there is a subfield $L$ of $K$ with $[K:L] = p$

Published on 26 Jun 2018 - 18:32

score 233 · Answer 6 · 2018-06-27 17:14:01

233

Views

Prove that if $ \operatorname{Gal}(K/F) \simeq \Bbb{Z}/2\Bbb{Z}\times\Bbb{Z}/2\mathbb{Z}$, then $K = F(\sqrt{a},\sqrt{b})$ for some $a,b \in F$

Published on 27 Jun 2018 - 17:14

score 139 · Answer 7 · 2018-06-28 21:58:10

139

Views

Show that $\displaystyle \min(F,a)=\prod_{1}^{r}(x-\tau_{i}(a))$

Published on 28 Jun 2018 - 21:58

score 431 · Answer 8 · 2018-06-29 03:41:26

431

Views

In a field extension, if each element's degree is bounded uniformly by $n$, is the extension finite?

Published on 29 Jun 2018 - 3:41

score 450 · Answer 9 · 2018-06-29 13:09:36

450

Views

Proving the number of K-automorphisms is bounded by the degree of the extension by considering field embeddings

Published on 29 Jun 2018 - 13:09

score 119 · Answer 10 · 2018-06-29 13:36:59

119

Views

The characteristic polynomial of $L_{a}$ is $\prod_{\sigma \in \mathrm{Gal}(K/F)}(x-\sigma(a))$ and the minimal polynomial of $L_{a}$ is $\min(F,a)$.

Published on 29 Jun 2018 - 13:36

score 257 · Answer 11 · 2026-03-24 23:46:14

257

Views

If $n$ divides $m$, prove that $\mathbb{Q}(\zeta_{n}) \subset \mathbb{Q}(\zeta_{m})$

Published on 24 Mar 2026 - 23:46

score 41 · Answer 12 · 2018-07-03 21:02:27

41

Views

Caracterization of algebraic extensions

Published on 03 Jul 2018 - 21:02

score 159 · Answer 13 · 2026-03-25 12:53:51

159

Views

How do I double the resolution of $2^{\nu_2(x)}$?

Published on 25 Mar 2026 - 12:53

score 52 · Answer 14 · 2018-07-06 08:33:28

52

Views

Is every finite dimensional field extension a matrix algebra?

Published on 06 Jul 2018 - 8:33

score 83 · Answer 15 · 2018-07-06 12:21:21

83

Views

Test for equality of quadratic extensions

Published on 06 Jul 2018 - 12:21

List Question

$|\mathrm{Gal}(L/K)| = 105$ and subgroups $A,B \leq \mathrm{Gal}(L/K)$ with $|A| = 21$ and $|B| = 35$. Prove that $L^{A}\cap L^{B} = K$.

Constructing a field in which polynomial has root

Show that every root of a polynomial is real (using Galois theory)

Determine $Gal(K/F)$ and find all the intermediate subfields of $K/F$.

Let $K$ be a Galois extension of $F$ with $[K/F] = n$. If $p$ is a prime divisor of $n$, show that there is a subfield $L$ of $K$ with $[K:L] = p$

Prove that if $ \operatorname{Gal}(K/F) \simeq \Bbb{Z}/2\Bbb{Z}\times\Bbb{Z}/2\mathbb{Z}$, then $K = F(\sqrt{a},\sqrt{b})$ for some $a,b \in F$

Show that $\displaystyle \min(F,a)=\prod_{1}^{r}(x-\tau_{i}(a))$

In a field extension, if each element's degree is bounded uniformly by $n$, is the extension finite?

Proving the number of K-automorphisms is bounded by the degree of the extension by considering field embeddings

The characteristic polynomial of $L_{a}$ is $\prod_{\sigma \in \mathrm{Gal}(K/F)}(x-\sigma(a))$ and the minimal polynomial of $L_{a}$ is $\min(F,a)$.

If $n$ divides $m$, prove that $\mathbb{Q}(\zeta_{n}) \subset \mathbb{Q}(\zeta_{m})$

Caracterization of algebraic extensions

How do I double the resolution of $2^{\nu_2(x)}$?

Is every finite dimensional field extension a matrix algebra?

Test for equality of quadratic extensions

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