Question

Definition of an extension field question

score 144 · Answer 1 · 2016-09-20 20:16:54

144

Views

Definition of an extension field question

Published on 20 Sep 2016 - 20:16

score 67 · Answer 2 · 2016-09-21 19:35:49

67

Views

Question about $\{1, \sqrt{3}\}$ being a basis of $\Bbb Q(\sqrt{3}, \sqrt{5})$ over $\Bbb Q(\sqrt{5})$

Published on 21 Sep 2016 - 19:35

score 131 · Answer 3 · 2016-09-23 14:57:56

131

Views

Question about determining $[\Bbb Q(\sqrt[3]2, \sqrt[4]3) : \Bbb Q]$

Published on 23 Sep 2016 - 14:57

score 789 · Answer 4 · 2016-09-24 15:49:07

789

Views

Separable closure and normality

Published on 24 Sep 2016 - 15:49

score 113 · Answer 5 · 2026-03-31 07:12:09

113

Views

If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$, given $f(x)$ and $g(x)$ are

Published on 31 Mar 2026 - 7:12

score 634 · Answer 6 · 2016-09-25 18:16:21

634

Views

Find a field with $a,b$ in an extension field such that $F(a, b) \neq F(a), F(a, b) \neq F(b)$, and $[F(a, b):F] \lt [F(a):F][F(b):F]$

Published on 25 Sep 2016 - 18:16

score 159 · Answer 7 · 2016-09-26 06:49:40

159

Views

Understanding the statement of a corollary from Dummit and Foote's Abstract Algebra third edition

Published on 26 Sep 2016 - 6:49

score 220 · Answer 8 · 2016-09-26 14:40:17

220

Views

If $[E_1:F]$ and $[E_2:F]$ are both prime, show that $E_1 = E_2$ or $E_1 \cap E_2 = F$

Published on 26 Sep 2016 - 14:40

score 565 · Answer 9 · 2016-09-26 16:34:32

565

Views

If $[E:F] = n$ and $p(x)$ is irreducible over $F$ and $\gcd(n, \deg p(x)) = 1$, then $[E(a) : E] \le [F(a) : F] = \deg f(x)$

Published on 26 Sep 2016 - 16:34

score 36 · Answer 10 · 2016-09-26 17:40:43

36

Views

$\text{dim} (L(\mathcal{S})|K(\mathcal{S}))\leq \text{dim} (L|K)$

Published on 26 Sep 2016 - 17:40

score 312 · Answer 11 · 2016-09-26 17:56:22

312

Views

Decomposing field extentions into algebraic and purely trancendental extentions

Published on 26 Sep 2016 - 17:56

score 1.5k · Answer 12 · 2026-03-27 00:43:33

1.5k

Views

Find a splitting field for $x^4 - x^2 - 2$ over $\Bbb Z_3$

Published on 27 Mar 2026 - 0:43

score 1.5k · Answer 13 · 2016-09-27 19:47:26

1.5k

Views

Let $a$ be a complex number that is algebraic over $\Bbb Q$ and let $r$ be a rational number. Prove that $a^r$ is algebraic over $\Bbb Q$.

Published on 27 Sep 2016 - 19:47

score 133 · Answer 14 · 2016-09-28 16:38:53

133

Views

Let $a$ be a generator of the group of nonzero elements of $GF(p^n)$ under multiplication. Then $a$ is algebraic over $GF( p)$ of degree $n$.

Published on 28 Sep 2016 - 16:38

score 160 · Answer 15 · 2016-09-30 14:25:53

160

Views

If $L_1$, $L_2$ are intermediate fields of $E/K$, then $[L_1 L_2 : K] = [L_1 : K] [L_2 : K] \implies L_1 \cap L_2 = K$.

Published on 30 Sep 2016 - 14:25

List Question

Definition of an extension field question

Question about $\{1, \sqrt{3}\}$ being a basis of $\Bbb Q(\sqrt{3}, \sqrt{5})$ over $\Bbb Q(\sqrt{5})$

Question about determining $[\Bbb Q(\sqrt[3]2, \sqrt[4]3) : \Bbb Q]$

Separable closure and normality

If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$, given $f(x)$ and $g(x)$ are

Find a field with $a,b$ in an extension field such that $F(a, b) \neq F(a), F(a, b) \neq F(b)$, and $[F(a, b):F] \lt [F(a):F][F(b):F]$

Understanding the statement of a corollary from Dummit and Foote's Abstract Algebra third edition

If $[E_1:F]$ and $[E_2:F]$ are both prime, show that $E_1 = E_2$ or $E_1 \cap E_2 = F$

If $[E:F] = n$ and $p(x)$ is irreducible over $F$ and $\gcd(n, \deg p(x)) = 1$, then $[E(a) : E] \le [F(a) : F] = \deg f(x)$

$\text{dim} (L(\mathcal{S})|K(\mathcal{S}))\leq \text{dim} (L|K)$

Decomposing field extentions into algebraic and purely trancendental extentions

Find a splitting field for $x^4 - x^2 - 2$ over $\Bbb Z_3$

Let $a$ be a complex number that is algebraic over $\Bbb Q$ and let $r$ be a rational number. Prove that $a^r$ is algebraic over $\Bbb Q$.

Let $a$ be a generator of the group of nonzero elements of $GF(p^n)$ under multiplication. Then $a$ is algebraic over $GF( p)$ of degree $n$.

If $L_1$, $L_2$ are intermediate fields of $E/K$, then $[L_1 L_2 : K] = [L_1 : K] [L_2 : K] \implies L_1 \cap L_2 = K$.

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