Question

Prove that $F(a_1, ... , a_i)$ contains no $p$th roots of unity not in $F(a_1, ... , a_{i-1})$

score 61 · Answer 1 · 2026-04-14 03:53:21

61

Views

Prove that $F(a_1, ... , a_i)$ contains no $p$th roots of unity not in $F(a_1, ... , a_{i-1})$

Published on 14 Apr 2026 - 3:53

score 31 · Answer 2 · 2026-04-16 19:50:50

31

Views

$O_{KL} \subset \frac{1}{d}O_KO_L$ where $d = \text{gcd}(disc(O_K),disc(O_L))$

Published on 16 Apr 2026 - 19:50

score 248 · Answer 3 · 2026-04-15 15:36:11

248

Views

An isomorphism of $F$ onto a subfield of $\bar{F}$ has the same number of extensions to each simple algebraic extension of $F$?.

Published on 15 Apr 2026 - 15:36

score 462 · Answer 4 · 2026-04-13 18:39:23

462

Views

$\mathbb{C}$ is isomorphic to $\bar{\mathbb{Q}}$?

Published on 13 Apr 2026 - 18:39

score 835 · Answer 5 · 2026-04-17 03:30:47

835

Views

Any algebraic closure of $\mathbb{Q}(\sqrt{2})$ is isomorphic to any algebraic closure of $\mathbb{Q}(\sqrt{17})$

Published on 17 Apr 2026 - 3:30

score 7.6k · Answer 6 · 2026-04-15 06:24:32

7.6k

Views

Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$

Published on 15 Apr 2026 - 6:24

score 47 · Answer 7 · 2026-04-17 03:00:54

47

Views

For the following polynomials (a) $f(x)=x^4-5x^2+6, F = \mathbb{Z}_7$, (b) $f(x)=x^3-3 , F=\mathbb{Q}$ in the given field $F$ find:

Published on 17 Apr 2026 - 3:00

score 2.3k · Answer 8 · 2026-04-14 13:42:29

2.3k

Views

Find the degree of the splitting field of $x^6+1$ over $\mathbb{Q}$

Published on 14 Apr 2026 - 13:42

score 1.3k · Answer 9 · 2026-04-15 09:39:23

1.3k

Views

Let $F$ be a field, and let $f(x)\in F[x]$ be a polynomial of prime degree. Suppose for every field extension $K$ of $F$

Published on 15 Apr 2026 - 9:39

score 222 · Answer 10 · 2026-04-11 23:15:13

222

Views

Field theory- Irreducible polynomial- Field extensions

Published on 11 Apr 2026 - 23:15

score 97 · Answer 11 · 2026-04-13 08:24:51

97

Views

Gauss Lucas Theorem over fields of positive characteristic

Published on 13 Apr 2026 - 8:24

score 226 · Answer 12 · 2026-04-12 09:00:55

226

Views

General method for finding the splitting field of $x^a+c$

Published on 12 Apr 2026 - 9:00

score 48 · Answer 13 · 2026-04-15 02:54:48

48

Views

Hint to prove that $x_1 ^2 + \cdots +x_k ^2 + 1$ has no roots in $\mathbb{Q}(\sqrt[3]{2}e^{\frac{2}{3}i \pi}) $

Published on 15 Apr 2026 - 2:54

score 198 · Answer 14 · 2026-04-15 15:43:09

198

Views

Why $f(\sigma(x))=\sigma(f(x))$ if $f$ is a polynomial with coefficients in $F$ and $\sigma$ is an automorphism that fixes $F$

Published on 15 Apr 2026 - 15:43

score 360 · Answer 15 · 2026-04-15 14:05:05

360

Views

Concept of field of roots vs decomposition field (splitting field): what is the difference?

Published on 15 Apr 2026 - 14:05

List Question

Prove that $F(a_1, ... , a_i)$ contains no $p$th roots of unity not in $F(a_1, ... , a_{i-1})$

$O_{KL} \subset \frac{1}{d}O_KO_L$ where $d = \text{gcd}(disc(O_K),disc(O_L))$

An isomorphism of $F$ onto a subfield of $\bar{F}$ has the same number of extensions to each simple algebraic extension of $F$?.

$\mathbb{C}$ is isomorphic to $\bar{\mathbb{Q}}$?

Any algebraic closure of $\mathbb{Q}(\sqrt{2})$ is isomorphic to any algebraic closure of $\mathbb{Q}(\sqrt{17})$

Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$

For the following polynomials (a) $f(x)=x^4-5x^2+6, F = \mathbb{Z}_7$, (b) $f(x)=x^3-3 , F=\mathbb{Q}$ in the given field $F$ find:

Find the degree of the splitting field of $x^6+1$ over $\mathbb{Q}$

Let $F$ be a field, and let $f(x)\in F[x]$ be a polynomial of prime degree. Suppose for every field extension $K$ of $F$

Field theory- Irreducible polynomial- Field extensions

Gauss Lucas Theorem over fields of positive characteristic

General method for finding the splitting field of $x^a+c$

Hint to prove that $x_1 ^2 + \cdots +x_k ^2 + 1$ has no roots in $\mathbb{Q}(\sqrt[3]{2}e^{\frac{2}{3}i \pi}) $

Why $f(\sigma(x))=\sigma(f(x))$ if $f$ is a polynomial with coefficients in $F$ and $\sigma$ is an automorphism that fixes $F$

Concept of field of roots vs decomposition field (splitting field): what is the difference?

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