# 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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Prove the following statements: a) The polynomial $x^3-10$ is irreducible over $\mathbb{Q}(\sqrt{2})$ b) The polynomial $x^3-10$ is irreducible over $\mathbb{Q}(\sqrt{-3})$ c) The polynomial $x^3-x-... 1answer 49 views ### Existence of irreducible polynomials with certain criteria Let$\mathbb{F}_{q}$be the finite field with$q$elements, where$q$is an odd prime power. The question is as follows: Does there exists$a\in \mathbb{F}_{q}^*\setminus (\mathbb{F}_{q}^*)^2$, such ... 2answers 29 views ### Is$x^3-1$reducible over$\mathbb{Q}$[duplicate] In other discussions, e.g. here it is claimed that a polynomial in a field$K$with degree greater than$1$, having a root in$K$, must be reducible. So by this criterion$X^3-1$would be reducible ... 0answers 18 views ### Is this proof of a irreducibility criterion in an integral domain correct? This is an exercise from Grillet's "Abstract Algebra" (page$145$, proposition$10.10$). Let$R$be an integral domain, let$I$be an ideal of$R$, and let$\pi\colon R\to R/I$be a canonical ... 4answers 818 views ### Show that$\mathbb{Q}(\sqrt2, \sqrt[3]2)$is a primitive field extension of$\mathbb{Q}$. I've tried a method similar to showing that$\mathbb{Q}(\sqrt2, \sqrt3)$is a primitive field extension, but the cube root of 2 just makes it a nightmare. Thanks in advance 1answer 74 views ### Irreducible Polynomial of Galois field We know that one irreducible polynomial on$\Bbb{Z}_2[x]$is$x^8 + x^4 + x^3 + x + 1$. How to check that it is irreducible? And how to generate irreducible polynomial for any degree? 2answers 42 views ###$f(x) = x$in$(\mathbb{Z}/6\mathbb{Z})[x]$factors as$(3x+4)(4x+3)$This question has a lot of parts so I'll post each part separately. First, show$f(x) = x$in$(\mathbb{Z}/6\mathbb{Z})[x]$factors as$(3x+4)(4x+3)$I am trying long division. I cannot divide$x$... 3answers 149 views ### Degree of splitting field of$X^4+2X^2+2$over$\mathbf{Q}$Find the degree of splitting field of$f=X^4+2X^2+2$over$\mathbf{Q}$. By Eisenstein,$f$is irreducible. By setting$Y=X^2$, we can solve for the roots:$Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$,$...
How do you find the units, irreducible elements and prime elements for $\mathbb{C}[𝑥]$, $\mathbb{R}[𝑥]$, $\mathbb{Q}[𝑥]$? Thank you.
### 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)(...