List questions from irreducible-polynomials

338

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$f_{p} (x^{p^{e-1}})$ is an irreducible polynomial in $\mathbb{Q}[x]$ for every prime $p$ and every postive integer$e$.

Published on 08 May 2020 - 22:12

124

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Help with Galois Groups

Published on 09 May 2020 - 3:26

167

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Rational Zero Test of Polynomials

Published on 25 Mar 2026 - 20:41

172

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If p (mod 4) = 3 and p is a Gaussian Prime. How to show that Z[i]/(p) is equal to GF(p^2)/(x^2+1)?

Published on 25 Mar 2026 - 12:49

256

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Doubt in IMO $1993$ Problem 1

Published on 12 May 2020 - 10:39

89

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Least $m$ such that $x^4+x^3+1$ divides $x^m-1$ over $\mathbb{F}_2$

Published on 12 May 2020 - 22:33

707

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Polynomial $x^3-2x^2-3x-4=0$

Published on 26 Mar 2026 - 22:17

137

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How to find an irreducible polynomial over a finite field with a primitive root (and low hamming weight)

Published on 26 Mar 2026 - 0:53

206

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Find an irreducible polynomial in $ Q[x]$ of degree $726$.

Published on 15 May 2020 - 11:28

49

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Irreducibility of polynomials degree $3$

Published on 30 Mar 2026 - 3:54

375

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$X^{p^k} - a ∈ K[X]$ irreducible?

Published on 25 Mar 2026 - 11:53

245

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Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $a, b, c, d∈ℤ$.

Published on 16 May 2020 - 19:21

58

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Is $(\mathbb{Z}/2\mathbb{Z})[T,T^{-1}]$ a DVR?

Published on 25 Mar 2026 - 10:53

68

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Isomorphic finite rings?

Published on 26 Mar 2026 - 9:38

77

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Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.

Published on 25 Mar 2026 - 11:54

List Question

$f_{p} (x^{p^{e-1}})$ is an irreducible polynomial in $\mathbb{Q}[x]$ for every prime $p$ and every postive integer$e$.

Help with Galois Groups

Rational Zero Test of Polynomials

If p (mod 4) = 3 and p is a Gaussian Prime. How to show that Z[i]/(p) is equal to GF(p^2)/(x^2+1)?

Doubt in IMO $1993$ Problem 1

Least $m$ such that $x^4+x^3+1$ divides $x^m-1$ over $\mathbb{F}_2$

Polynomial $x^3-2x^2-3x-4=0$

How to find an irreducible polynomial over a finite field with a primitive root (and low hamming weight)

Find an irreducible polynomial in $ Q[x]$ of degree $726$.

Irreducibility of polynomials degree $3$

$X^{p^k} - a ∈ K[X]$ irreducible?

Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $a, b, c, d∈ℤ$.

Is $(\mathbb{Z}/2\mathbb{Z})[T,T^{-1}]$ a DVR?

Isomorphic finite rings?

Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.

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