Question

To prove $f=x^4+x^3+x^2+x^1+1$ is irreducible over the $\mathbb{Q}$

score 202 · Answer 1 · 2021-07-28 19:46:58

202

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To prove $f=x^4+x^3+x^2+x^1+1$ is irreducible over the $\mathbb{Q}$

Published on 28 Jul 2021 - 19:46

score 79 · Answer 2 · 2026-04-10 22:49:23

79

Views

Every polynomial with constant term 1 can be factorized using degree one polynomials of the for (1-ax)

Published on 10 Apr 2026 - 22:49

score 96 · Answer 3 · 2021-08-01 00:12:30

96

Views

Is $\mathbb{Q}(\alpha^8)=\mathbb{Q}(\alpha)$?

Published on 01 Aug 2021 - 0:12

score 521 · Answer 4 · 2021-08-02 06:31:44

521

Views

Irreducibilty of polynomial $x^9-6x^6+282x^3-8$ over $\mathbb {Q} $

Published on 02 Aug 2021 - 6:31

score 64 · Answer 5 · 2021-08-03 16:38:53

64

Views

Let $f (x)$ and $g(x)$ be nonzero polynomials in $Q[x]$. Consider the gcds of $f (x)$ and $g(x)$ in $Q[x]$, $R[x]$ and $C[x]$. Are the GCDs same?

Published on 03 Aug 2021 - 16:38

score 192 · Answer 6 · 2026-04-13 08:04:25

192

Views

Two different degrees for a ODE

Published on 13 Apr 2026 - 8:04

score 220 · Answer 7 · 2021-08-17 16:42:26

220

Views

Factor $x^{17}+1$ into a product of irreducibles

Published on 17 Aug 2021 - 16:42

score 274 · Answer 8 · 2021-08-21 12:26:29

274

Views

Let $f$ be a monic polynomial in $\mathbb{Z}[x]$. Request to verify my thinking about a question on irreducibility of $f$.

Published on 21 Aug 2021 - 12:26

score 42 · Answer 9 · 2026-04-05 14:56:22

42

Views

For $m\geq2$, is $\Bbb{F}[x]/(p(x)^m)$ a field?

Published on 05 Apr 2026 - 14:56

score 64 · Answer 10 · 2026-04-10 20:56:36

64

Views

Polynomial factorisation problem

Published on 10 Apr 2026 - 20:56

score 195 · Answer 11 · 2026-04-12 19:28:21

195

Views

Can the root of an irreducible polynomial be the root of a lower degree polynomial as well?

Published on 12 Apr 2026 - 19:28

score 86 · Answer 12 · 2026-04-03 02:00:56

86

Views

Let $F$ be a field and $E$ a split field of irred poly $f(x)\in F[x]$. Show that if $c,d\in F$ and $c\neq 0$ then poly $f(cx+d)$ splits in $E[x]$.

Published on 03 Apr 2026 - 2:00

score 423 · Answer 13 · 2021-08-27 01:22:57

423

Views

Suppose that f(x) and g(x) are irreducible over F and that deg f(x) and deg g(x) are relatively prime. Show g(x) is irreducible over F(a)

Published on 27 Aug 2021 - 1:22

score 600 · Answer 14 · 2026-04-13 07:55:47

600

Views

Absolute irreducibility of a polynomial over a finite field

Published on 13 Apr 2026 - 7:55

score 87 · Answer 15 · 2021-08-31 15:01:07

87

Views

Minimal Polynomials and Limiting Properties of the $n$-bonacci Constants.

Published on 31 Aug 2021 - 15:01

List Question

To prove $f=x^4+x^3+x^2+x^1+1$ is irreducible over the $\mathbb{Q}$

Every polynomial with constant term 1 can be factorized using degree one polynomials of the for (1-ax)

Is $\mathbb{Q}(\alpha^8)=\mathbb{Q}(\alpha)$?

Irreducibilty of polynomial $x^9-6x^6+282x^3-8$ over $\mathbb {Q} $

Let $f (x)$ and $g(x)$ be nonzero polynomials in $Q[x]$. Consider the gcds of $f (x)$ and $g(x)$ in $Q[x]$, $R[x]$ and $C[x]$. Are the GCDs same?

Two different degrees for a ODE

Factor $x^{17}+1$ into a product of irreducibles

Let $f$ be a monic polynomial in $\mathbb{Z}[x]$. Request to verify my thinking about a question on irreducibility of $f$.

For $m\geq2$, is $\Bbb{F}[x]/(p(x)^m)$ a field?

Polynomial factorisation problem

Can the root of an irreducible polynomial be the root of a lower degree polynomial as well?

Let $F$ be a field and $E$ a split field of irred poly $f(x)\in F[x]$. Show that if $c,d\in F$ and $c\neq 0$ then poly $f(cx+d)$ splits in $E[x]$.

Suppose that f(x) and g(x) are irreducible over F and that deg f(x) and deg g(x) are relatively prime. Show g(x) is irreducible over F(a)

Absolute irreducibility of a polynomial over a finite field

Minimal Polynomials and Limiting Properties of the $n$-bonacci Constants.

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