# Questions tagged [irreducible-polynomials]

Often called prime polynomials. Polynomials that have no polynomial divisors.

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### Galois group of $x^4+tx+t\in F(t)[x]$

Let $F$ be a field of characteristic $\operatorname{char}(F)\ne2$. Prove that the polynomial $f(x)=x^4+tx+t\in F(t)[x]$ is irreduciable, and find its Galois group. Proving that the polynomial is ...
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### Is there any website like OEIS for special polynomials

I would like to know if there is any kind of website (like OEIS) in which we can search for special known polynomials. For example, we put the coefficients of Legendre's and then the website gives us ...
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### All irreducible polynomials of degree 4 over a field of 4 elements [closed]

Find out all irreducible polynomials of degree 4 over a field of 4 elements. Help me, please
### Let $F=\Bbb{F}_3[x]/(f(x))$ and $\alpha=x+f(x) \Bbb{F}_3 \in F$, so that $f(\alpha)=0$ Prove that f(x) is irreducible
Let $f(x)=x^3+2x^2+1 \in \Bbb{F}_3[x]$. Let $F=\Bbb{F}_3[x]/(f(x))$ and $\alpha=x+f(x) \Bbb{F}_3 \in F$, so that $f(\alpha)=0$ Prove that f(x) is irreducible. Gauss' lemma: suppose $f \in \Bbb{Z}[x]$ ...
### Prove polynomial is irreducible in $Z_p[x]$.
Give $f(x)=a_0+a_1x+\ldots+a_nx^n$ and prime $p$ that $p \nmid a_n$ and $GCD(a_1,a_2,\ldots,a_n)=1$. Which one in two clause below is correct? (1):"If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then \$f(...