Question

Show $\mathbb Z[x] / I \cong \overline{\mathbb Z}[x]/ \overline{I}$ where $\overline{\mathbb Z} = (\mathbb Z /3 \mathbb Z)$

score 83 · Answer 1 · 2019-07-13 13:23:46

83

Views

Show $\mathbb Z[x] / I \cong \overline{\mathbb Z}[x]/ \overline{I}$ where $\overline{\mathbb Z} = (\mathbb Z /3 \mathbb Z)$

Published on 13 Jul 2019 - 13:23

score 234 · Answer 2 · 2019-07-14 14:29:40

234

Views

If $E/F$ is a field extension of degree $2$ and $\text{char}(F)\ne 2$ then $E=F(\sqrt a)$ for some $a\in E$

Published on 14 Jul 2019 - 14:29

score 48 · Answer 3 · 2019-07-14 16:34:45

48

Views

If $E/F$ is a field extension with $a,b\in E$ of "degree" $n,m$ then $ab$ and $a+b$ have "degree" less then $mn$

Published on 14 Jul 2019 - 16:34

score 75 · Answer 4 · 2019-07-18 09:01:15

75

Views

Factor $f(x) = x^5 - 2x^4 - x^3 + 2x^2 -2x + 4 \in \mathbb{Q}[x]$ into a product of irreducible polynomials in $\mathbb{Q},\mathbb{R},\mathbb{C}$

Published on 18 Jul 2019 - 9:01

score 30 · Answer 5 · 2019-07-18 14:29:44

30

Views

Under what conditions $[E:F]=\deg(f)$ where $f\in F[x]$ and $E\setminus F$ is a field extension

Published on 18 Jul 2019 - 14:29

score 69 · Answer 6 · 2019-07-19 00:44:06

69

Views

Factor $g(x) = x^3 - x^2 -x - \overline{2} \in \mathbb{Z}_5$ into a product of irreducible polynomials in $\mathbb{Z}_5$

Published on 19 Jul 2019 - 0:44

score 54 · Answer 7 · 2019-07-19 00:49:35

54

Views

Determine polynomial of $3$ degree real coefficients $f(x)$ such that $f(x) \vdots (x-2)$ and $f(x)$ dividing by $x^2-1$ have remainder $2x$

Published on 19 Jul 2019 - 0:49

score 117 · Answer 8 · 2019-07-20 18:41:20

117

Views

Showing $x^4-16x^2+4$ is irreducible over $\mathbb{Q}$

Published on 20 Jul 2019 - 18:41

score 254 · Answer 9 · 2019-07-22 06:59:10

254

Views

$\frac{\mathbb{C[x]}}{\langle x-a\rangle}$ is isomorphic to which field?

Published on 22 Jul 2019 - 6:59

score 263 · Answer 10 · 2026-03-27 04:38:36

263

Views

When is $y=3x^2+3x+1$ a prime number in $\mathbb{Z}$ with $x \in \mathbb{Z}$?

Published on 27 Mar 2026 - 4:38

score 447 · Answer 11 · 2019-07-23 17:32:50

447

Views

Prove that there exists an irreducible polynomial $f\in\mathbb{F}_p[x]$ of degree $2014$

Published on 23 Jul 2019 - 17:32

score 226 · Answer 12 · 2019-07-26 07:41:41

226

Views

Irreducible polynomial of degree 3 over rational numbers is separable or not

Published on 26 Jul 2019 - 7:41

score 367 · Answer 13 · 2026-03-26 01:28:52

367

Views

What do the elements of $\mathbb{F}_{243}$ look like?

Published on 26 Mar 2026 - 1:28

score 122 · Answer 14 · 2019-07-27 01:35:52

122

Views

If $f$ is not primitive, then it is not irreducible?

Published on 27 Jul 2019 - 1:35

score 63 · Answer 15 · 2019-07-28 09:08:26

63

Views

What does the 3rd volume of Lehrbuch der Algebra of Heinrich Weber talk about?

Published on 28 Jul 2019 - 9:08

List Question

Show $\mathbb Z[x] / I \cong \overline{\mathbb Z}[x]/ \overline{I}$ where $\overline{\mathbb Z} = (\mathbb Z /3 \mathbb Z)$

If $E/F$ is a field extension of degree $2$ and $\text{char}(F)\ne 2$ then $E=F(\sqrt a)$ for some $a\in E$

If $E/F$ is a field extension with $a,b\in E$ of "degree" $n,m$ then $ab$ and $a+b$ have "degree" less then $mn$

Factor $f(x) = x^5 - 2x^4 - x^3 + 2x^2 -2x + 4 \in \mathbb{Q}[x]$ into a product of irreducible polynomials in $\mathbb{Q},\mathbb{R},\mathbb{C}$

Under what conditions $[E:F]=\deg(f)$ where $f\in F[x]$ and $E\setminus F$ is a field extension

Factor $g(x) = x^3 - x^2 -x - \overline{2} \in \mathbb{Z}_5$ into a product of irreducible polynomials in $\mathbb{Z}_5$

Determine polynomial of $3$ degree real coefficients $f(x)$ such that $f(x) \vdots (x-2)$ and $f(x)$ dividing by $x^2-1$ have remainder $2x$

Showing $x^4-16x^2+4$ is irreducible over $\mathbb{Q}$

$\frac{\mathbb{C[x]}}{\langle x-a\rangle}$ is isomorphic to which field?

When is $y=3x^2+3x+1$ a prime number in $\mathbb{Z}$ with $x \in \mathbb{Z}$?

Prove that there exists an irreducible polynomial $f\in\mathbb{F}_p[x]$ of degree $2014$

Irreducible polynomial of degree 3 over rational numbers is separable or not

What do the elements of $\mathbb{F}_{243}$ look like?

If $f$ is not primitive, then it is not irreducible?

What does the 3rd volume of Lehrbuch der Algebra of Heinrich Weber talk about?

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