# Questions tagged [irreducible-polynomials]

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

2,063 questions
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### Irreducibility of $x^n-a^n$ on $F(a^n)$ where $a$ is transcendental on $F$

I'm looking for a short proof of the following : If $F$ is a field where $a$ is transcendental, $x^n-a^n$ is irreducible on $F(a^n)$. I've managed to prove it in a tedious way but I feel like there ...
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### Why $F_2[X]/(X^2+X+1)$ has $4$ elements and what are those?

I don't understand the three claims that some $F_a[X]/(p(x))$ has some $n$ elements in the following text (from Adkins' Algebra): For example Why $F_2[X]/(X^2+X+1)$ has $4$ elements and what are ...
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### Show $x^{p}+a$ is reducible in $\mathbb{Z}_p[x]$

first time taking abstract algebra stuck on this question and have no idea how to go about it. Show that $x^{p}+a$ is reducible in $\mathbb{Z_{p}}[x]$: for each $a$ in $\mathbb{Z_{p}}$, where $p$ is a ...
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### Why we ignore the other elements of $F_{p^k}$ when we check if any element of it is a root of multiplicity greater than $1$ in $p(x)$ or not?

$p(x) = x^4 + x +6$ . I was to find if there is any root of multiplicity greater than $1$ in the field of characteristics $p$. I was suggested to check every element of the $F_p$. But a filed ...
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### How to prove this polynomial is irreducible over $\mathbb{Q}$

$f(x)＝x^{n}+x+p$ where $p\geq 3$ is a prime.Prove that $f(x)$ is irreducible over $\mathbb{Q}$. My try:Let $y＝x+m$ for some $m$ and try to use Eisenstein's Criterion.But I can't find the suitable $m$....
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### $y^2z - x^3$ is irreducible and with one singularity

Give an example for an irreducible cubic curve in $\mathbb{C}\mathbb{P}^2$ with exactly one singular point. It is easy to check that $y^2z - x^3$ has only [0,0,1] as a singularity. But how to show it ...
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### Irreducibility of $X^{p-1} + \cdots + X+1$

Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with $X+1$, ...
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### Show some polynomial satisfies Eisenstein's Criterion

Consider a polynomial $$f(X) = X^{(p-1)p^{n-1}} + X^{(p-2)p^{n-1}} + \cdots + X^{p^{n-1}} + 1$$ Now I need to show $f(X+1)$ safeties Eisenstein's criterion. My argument is that f(X+1) = (X+1)^{...
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### Units, Primes and Irreducibles

How do you find the units, irreducible elements and prime elements for $\mathbb{C}[𝑥]$, $\mathbb{R}[𝑥]$, $\mathbb{Q}[𝑥]$? Thank you.
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### Degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$

What is the degree of $\cos(2\pi/8) + i \sin(2\pi/8)$ over $\mathbb{Q}$ ? I note that $\cos(2\pi/8) + i \sin(2\pi/8)$ is a root of $x^8-1$. $x^8-1$ can be factored into \$x^8-1 =(x^4+1)(x^2+1)(x+1)(...