Question

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

score 52 · Answer 1 · 2016-06-14 18:13:38

52

Views

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

Published on 14 Jun 2016 - 18:13

score 74 · Answer 2 · 2016-06-15 00:18:20

74

Views

Reducibility of $x^q -x -1$ in $\mathbb{F}_{q}$

Published on 15 Jun 2016 - 0:18

score 220 · Answer 3 · 2016-06-15 05:34:56

220

Views

Irreducible polynomial of every degree over finite field

Published on 15 Jun 2016 - 5:34

score 990 · Answer 4 · 2026-04-04 12:07:29

990

Views

Ideal generated by two irreducible polynomials is the field itself

Published on 04 Apr 2026 - 12:07

score 101 · Answer 5 · 2016-06-16 10:32:30

101

Views

Irreducible over $\mathbb{Q}$ (ring $\mathbb{Z}$)

Published on 16 Jun 2016 - 10:32

score 2.2k · Answer 6 · 2016-06-17 19:22:39

2.2k

Views

If $q(X)$ is reducible in $\mathbb Z[X]$, then it's reducible in $\mathbb Z_p[X]$ for every prime $p$

Published on 17 Jun 2016 - 19:22

score 252 · Answer 7 · 2016-06-18 01:14:15

252

Views

Minimal polynomial for $x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$

Published on 18 Jun 2016 - 1:14

score 226 · Answer 8 · 2016-06-19 15:12:16

226

Views

If $a$ is algebraic, prove that there is a minimal polynomial $p(x)$ in $Q[x]$ such $p(a)$ = $0$.

Published on 19 Jun 2016 - 15:12

score 36 · Answer 9 · 2016-06-19 17:41:35

If $a$ is algebraic and $f\colon\mathbb{Q}[x]\to\mathbb{C}$ where $f(g(x))=g(a)$, prove that $\ker(f)$ is a maximal ideal of $\mathbb{Q}[x]$

score 46 · Answer 10 · 2016-06-20 12:52:08

46

Views

Irreducible polynomial on $\mathbb{Z}_2$-field

Published on 20 Jun 2016 - 12:52

score 579 · Answer 11 · 2016-06-20 19:14:15

579

Views

Number of even irreducible monic polynomials of a given degree over a finite field

Published on 20 Jun 2016 - 19:14

score 212 · Answer 12 · 2026-04-02 16:55:42

212

Views

Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Published on 02 Apr 2026 - 16:55

score 295 · Answer 13 · 2016-06-21 10:16:43

295

Views

Show that $\mathbb{F}_9 \not \subset \mathbb{F}_{27}$

Published on 21 Jun 2016 - 10:16

score 115 · Answer 14 · 2026-04-04 17:33:21

115

Views

Prove that $q(x)$ does not divide $p(x)$?

Published on 04 Apr 2026 - 17:33

score 64 · Answer 15 · 2016-06-22 14:03:16

64

Views

Construction of field extension for $[E:\mathbb F_{11}]=3$

Published on 22 Jun 2016 - 14:03

List Question

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

Reducibility of $x^q -x -1$ in $\mathbb{F}_{q}$

Irreducible polynomial of every degree over finite field

Ideal generated by two irreducible polynomials is the field itself

Irreducible over $\mathbb{Q}$ (ring $\mathbb{Z}$)

If $q(X)$ is reducible in $\mathbb Z[X]$, then it's reducible in $\mathbb Z_p[X]$ for every prime $p$

Minimal polynomial for $x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$

If $a$ is algebraic, prove that there is a minimal polynomial $p(x)$ in $Q[x]$ such $p(a)$ = $0$.

If $a$ is algebraic and $f\colon\mathbb{Q}[x]\to\mathbb{C}$ where $f(g(x))=g(a)$, prove that $\ker(f)$ is a maximal ideal of $\mathbb{Q}[x]$

Irreducible polynomial on $\mathbb{Z}_2$-field

Number of even irreducible monic polynomials of a given degree over a finite field

Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Show that $\mathbb{F}_9 \not \subset \mathbb{F}_{27}$

Prove that $q(x)$ does not divide $p(x)$?

Construction of field extension for $[E:\mathbb F_{11}]=3$

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