# Questions tagged [irreducible-polynomials]

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

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### Is the area enclosed by p(x,y) always irrational?

Take a polynomial $p \in \mathbb{Q}[X,Y]$. Now draw the graph of $p(x,y)=0$. If, like $X^2-Y^2-1$, this turns out to enclose a finite area, is the area enclosed always irrational? There are some ...
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### Prove that $x^6+5x^2+8$ is reducible over Z (integer)?

$attempts:-$ 1] I tried to replace $X^2=t$ but nothing click after that . 2] then I tried to replace this polynomial say P(x) by P(x+1) or P(x-1) to apply Eisenstein's Irreducibility Criterion Theorem ...
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### Proposition 8 Corollary 1, Section 5.7 of Hungerford’s Algebra

Corollary 1.9. Let $E$ and $F$ each be extension fields of $K$ and let $u\in E$ and $v\in F$ be algebraic over $K$. Then $u$ and $v$ are roots of the same irreducible polynomial $f \in K[x]$ if and ...
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### Polynomial reduction modulo n. Irreducible polynomal.

I have the following polynomial: $f(x)=x^4+1$. I have to prove that it is irreducible over $\mathbb{Z}[x]$ using reduction criterion. The Reduction Criterion says that: Let $\mathfrak{m}$ be maximal ...
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### Irreducibility of the $p^k$-th cyclotomic polynomial

I want to prove that the cyclotomic polynomial $\Phi_{p^k}$ is irreducible using Eisenstein (I know that every cyclotomic polynomial is irreducible, I am just trying this approach). I am exposing what ...
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### To determine the number of roots for all antiderivative of a cubic polynomial

Let $f(x)$ be a cubic polynomial with real coefficients. Suppose that $f(x)$ has exactly one real root which is simple. Which of the following statements holds for all antiderivative $F(x)$ of $f(x)$ ?...
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### An efficient algorithm for determining whether a quartic with integer coefficients is irreducible over $\mathbb{Z}$

I'm interested in what efficient algorithm could be used for determining if a quartic polynomial with integer coefficients is irreducible over $\mathbb{Z}$. For quadratics and cubics it's not too bad, ...
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### If $f(x)\in \mathbb{Z}[x]$ is irreducible (over $\mathbb{Q}$), is it always possible to find $a$ and $b$ in $\mathbb{Q}$ with $f(ax+b)$ Eisenstein? [duplicate]

My initial thought is no, simply because it seems too easy if it is true. The simplest example of a nontrivial irreducible polynomial I could think of was $f(x)=x^2+1$. Unfortunately, $f(x+1)$ is ...
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### Finding the irreducible elements of a polynomial ring

Let $R = \{f(X)\in \mathbb{Q}[X] \space | \space f(0)\in \mathbb{Z}\}$. I am asked to find the irreducible elements of $R$. I have found that the units in $R$ are $\pm 1$. If $\deg f = 0$, i.e. $f$ is ...
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### $X^{6}+(T-22) X^{4}+T X^{3}+(T-22) X^{2}+1$ is irreducible over $\mathbb Q(T)$

I'm asked to prove that $f(X)=X^{6}+(T-22) X^{4}+T X^{3}+(T-22) X^{2}+1$ is irreducible over $\mathbb Q(T)$. For this, I need first to prove that the polynomials $p(X) = X^{6}-22 X^{4}-22 X^{2}+1$ and ...
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### For what integers $m\gt n\gt 0$, the polynomial $x^m+x^n+1$ is irreducible over $\mathbb Q$?

I came up with this problem and have found it interesting. Problem. For what integers $m\gt n\gt 0$, the polynomial $f(x)=x^m+x^n+1$ is irreducible in $\mathbb Q[x]$? If $mn\equiv 2 \pmod 3$, i.e. one ...
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### Irreducibility over Field extensions.

This is part of something bigger that I'm trying to prove, but I'm having difficulties with this part, which seems like it should be relatively simple. The polynomial $(x^2 - p)$, $p$ prime, is ...
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### Algebraic set of irreducible polynomial is irreducible?

I'm going through Fulton's Algebraic Curves, and just completed an exercise showing that $V(Y-X^2)$ is irreducible. My solution for this was to show that $Y-X^2$ is irreducible in $k[X,Y]$ (by showing ...
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### Irreducibility of $X^4-\sqrt{2}$ over $\mathbb{Q}(\sqrt{2})$.

To prove that $X^4-23$ is irreducible in $\mathbb{Q}[X]$ we can do the following: We use the Eisenstein criterion with $a=23$ to see that it is irreducible in $\mathbb{Z}[X]$, and then we conclude ...
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### $f$ and $Df$ are not relative primes in $F[X]$, then they are not relative primes in $K[X]$.

The question I will ask originates in the context of the theory of perfect fields and separable extensions, but it is a question of irreducibility of polynomials between extensions of fields and of ...
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### is $x^4-x+1$ irreducible in $\mathbb{Z}_3$
i was wondering if i checked correctly. i found all polynomials in $\mathbb{Z}_3[x]$ of degree 2 which are irreducible and checked if they are divisible without remainder the polynomials i tried were \$...