Question

Let $\mathbb{R} \supset \mathbb{Q}$ and $\alpha = 2 + \sqrt[3]{4}$. Find the minimal polynomial of $\alpha$.

score 68 · Answer 1 · 2020-01-08 11:50:16

68

Views

Let $\mathbb{R} \supset \mathbb{Q}$ and $\alpha = 2 + \sqrt[3]{4}$. Find the minimal polynomial of $\alpha$.

Published on 08 Jan 2020 - 11:50

score 48 · Answer 2 · 2020-01-08 12:19:46

48

Views

Show that $f(x)$ is irreducible and if $a \in \mathbb{C}$ is a root of $f(x)$ then $\frac{-1}{a+1}$ is also a root of $f(x)$

Published on 08 Jan 2020 - 12:19

score 39 · Answer 3 · 2020-01-08 13:03:14

39

Views

Show that $[E:k] \le n!$

Published on 08 Jan 2020 - 13:03

score 72 · Answer 4 · 2026-03-28 11:22:07

72

Views

Calculate $[E: \mathbb{Q}]$ and show that $E$ is a Galois extension on $\mathbb{Q}$

Published on 28 Mar 2026 - 11:22

score 48 · Answer 5 · 2020-01-08 17:34:15

48

Views

Showing that an extension is normal

Published on 08 Jan 2020 - 17:34

score 76 · Answer 6 · 2026-03-28 11:21:45

76

Views

Show that $F$ is a Galois extension on $\mathbb{Z}_p$ and Show that for each $a \in F$ then $a = x^2 + y^2$ with $x,y \in F$

Published on 28 Mar 2026 - 11:21

score 56 · Answer 7 · 2020-01-08 18:35:29

56

Views

Multiplicity of Zeroes and Derivatives

Published on 08 Jan 2020 - 18:35

score 28 · Answer 8 · 2020-01-09 04:32:40

Prove that if extensions $E_1$ and $E_2$ are isomorphic over $F$, $u$ is a root of $f(x)$ over $E_1$, then $\sigma(u)$ is a root over $E_2$

score 85 · Answer 9 · 2026-03-25 17:40:38

85

Views

On the space of polynomials in $T$ (a linear operator) with no non-zero invariant subspace

Published on 25 Mar 2026 - 17:40

score 186 · Answer 10 · 2020-01-09 17:17:38

186

Views

Generic points and smoothness of schemes

Published on 09 Jan 2020 - 17:17

score 117 · Answer 11 · 2020-01-10 00:55:45

117

Views

If $\alpha$ is transcendental, why is the evaluation homomorphism $k[t] \to k(\alpha)$ not onto? What does a general element in $k(\alpha)$ look like?

Published on 10 Jan 2020 - 0:55

score 900 · Answer 12 · 2020-01-10 23:15:18

900

Views

Understanding field extension

Published on 10 Jan 2020 - 23:15

score 84 · Answer 13 · 2020-01-11 08:27:16

84

Views

Another product in $\mathbb{R}^2$

Published on 11 Jan 2020 - 8:27

score 170 · Answer 14 · 2020-01-11 12:09:40

170

Views

Computing index of unit groups of rings of integers without explicitely computing them

Published on 11 Jan 2020 - 12:09

score 549 · Answer 15 · 2020-01-12 19:20:18

549

Views

Polynomial whose roots adjoin to a field $\mathbb Q(\sqrt6)$ to give an extension field $\mathbb Q(\sqrt2,\sqrt3)$

Published on 12 Jan 2020 - 19:20

List Question

Let $\mathbb{R} \supset \mathbb{Q}$ and $\alpha = 2 + \sqrt[3]{4}$. Find the minimal polynomial of $\alpha$.

Show that $f(x)$ is irreducible and if $a \in \mathbb{C}$ is a root of $f(x)$ then $\frac{-1}{a+1}$ is also a root of $f(x)$

Show that $[E:k] \le n!$

Calculate $[E: \mathbb{Q}]$ and show that $E$ is a Galois extension on $\mathbb{Q}$

Showing that an extension is normal

Show that $F$ is a Galois extension on $\mathbb{Z}_p$ and Show that for each $a \in F$ then $a = x^2 + y^2$ with $x,y \in F$

Multiplicity of Zeroes and Derivatives

Prove that if extensions $E_1$ and $E_2$ are isomorphic over $F$, $u$ is a root of $f(x)$ over $E_1$, then $\sigma(u)$ is a root over $E_2$

On the space of polynomials in $T$ (a linear operator) with no non-zero invariant subspace

Generic points and smoothness of schemes

If $\alpha$ is transcendental, why is the evaluation homomorphism $k[t] \to k(\alpha)$ not onto? What does a general element in $k(\alpha)$ look like?

Understanding field extension

Another product in $\mathbb{R}^2$

Computing index of unit groups of rings of integers without explicitely computing them

Polynomial whose roots adjoin to a field $\mathbb Q(\sqrt6)$ to give an extension field $\mathbb Q(\sqrt2,\sqrt3)$

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