List questions from irreducible-polynomials

Published on 07 Apr 2015 - 21:27

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Prove that the following cubic polynomial is irreducible over $\mathbb Q$

Published on 07 Apr 2015 - 22:32

686

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Example of Field Extension $E/F$ with $Char(F)=2$ and $[E:F]=2$, but is not Galois

Published on 07 Apr 2015 - 23:41

89

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Proving $f(x)=x^n-p$ is minimal of $\alpha=\sqrt[n]{p}$ over the field F (p is prime)

Published on 09 Apr 2015 - 22:22

61

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Question about ideals

Published on 02 Apr 2026 - 23:54

681

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How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

Published on 11 Apr 2015 - 12:57

248

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$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

Published on 12 Apr 2015 - 12:10

27

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Is this condition enough for irreducibility?

Published on 14 Apr 2015 - 14:10

1.2k

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Galois group of $f$ is cyclic if $\deg f$ is prime

Published on 18 Apr 2015 - 19:28

106

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Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

Published on 18 Apr 2015 - 20:47

373

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Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

Published on 19 Apr 2015 - 10:54

63

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Polynomial with degree less than degree of an irreducible polynomial of the same root is 0

Published on 20 Apr 2015 - 18:43

99

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Show that $\bar{a}_{n}(\bar{x})^n+···+\bar{a}_{1}\bar{x}+\bar{a}_{0}=0_{F[x]/I}$

Published on 03 Apr 2026 - 1:37

941

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Techniques to prove that two field extensions are distinct

Published on 22 Apr 2015 - 10:26

47

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Power of a root in a field

Published on 22 Apr 2015 - 18:34

List Question

General polynomial of degree $n$ is irreducible from Gauss' Lemma

Prove that the following cubic polynomial is irreducible over $\mathbb Q$

Example of Field Extension $E/F$ with $Char(F)=2$ and $[E:F]=2$, but is not Galois

Proving $f(x)=x^n-p$ is minimal of $\alpha=\sqrt[n]{p}$ over the field F (p is prime)

Question about ideals

How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

Is this condition enough for irreducibility?

Galois group of $f$ is cyclic if $\deg f$ is prime

Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

Polynomial with degree less than degree of an irreducible polynomial of the same root is 0

Show that $\bar{a}_{n}(\bar{x})^n+···+\bar{a}_{1}\bar{x}+\bar{a}_{0}=0_{F[x]/I}$

Techniques to prove that two field extensions are distinct

Power of a root in a field

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