# Questions tagged [irreducible-polynomials]

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

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### A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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### How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not ...
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### Is $f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$?

In this document on page $3$ I found an interesting polynomial: $$f_n=\frac{(x+1)^n-(x^n+1)}{x}.$$ Question is whether this polynomial is irreducible over $\mathbf{Q}$ for arbitrary $n \geq 1$. ...
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### Number of monic irreducible polynomials of prime degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$? Thanks!
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### Irreducible but not prime element

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any ...
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### Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem I....
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### Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
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### How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't help ...
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### Prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$

How to prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$? I tried Eisenstein criteria on $f(x+n)$ with $n$ ranging from $-10$ to $10$. None can be applied. I tried ...
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### Are derivatives of geometric progressions all irreducible?

Consider the polynomials $P_n(x)=1+2x+3x^2+\dots+nx^{n-1}$. Problem A5 in 2014 Putnam competition was to prove that these polynomials are pairwise relatively prime. In the solution sheet there is the ...
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### When is $X^n-a$ is irreducible over F?

Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show ...
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### Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$

I want to prove that for any set of distinct integers $a_1,\ldots,a_n$, the polynomial $$h = (x-a_1)\cdots(x-a_n) + 1$$ is irreducible over the field $\mathbb{Q}$, except for the following special ...
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### Reducibility over $\mathbb{Z}_2$?

I have seen that $x^{2}+x+1$ and $x^{4}+x+1$ are irreducible over $\mathbb{Z}_2$ and I thought a polynomial of the form $x^{2^m}+x+1$ for $m\ge3$ would be irreducible too. However using WolframAlpha, ...
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### $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$

I am trying to show that $X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$ where $n,m$ are positive integers. I showed that if $\gcd(n,m)$ is not $1$, then $X^n-Y^m$ is reducible. How to ...
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### How prove $P(x)=\sum_{k=0}^{n}(2k+1)x^k$ is irreducible over $\mathbb{Q}$

Show that the polynomial $$P(x)=\sum_{k=0}^{n}(2k+1)x^k,\forall n\in N^{+}$$ is irreducible over $Q$. My try: Since $P(x)$ has integer coefficients and the gcd of these coefficients is $1$, by Gauss'...
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### $x^2 +y^2 + z^2$ is irreducible in $\mathbb C [x,y,z]$

Is $x^2 +y^2 + z^2$ irreducible in $\mathbb C [x,y,z]$? As $(x^2+y^2+z^2)= (x+y+z)^2- 2(xy+yz+zx)$, $$(x^2+y^2+z^2)=\left(x+y+z+\sqrt{2(xy+yz+zx)}\right)\left(x+y+z-\sqrt{2(xy+yz+zx)}\right).$$ But ...
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### Show that $x^4 + 8$ is irreducible over Z

Is there an easy way to show that $x^4+8$ is irreducible over $\mathbb Z$ without trying to write it as a product of polynomials of lower degrees?
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### Irreducibility of a polynomial if it has no root (Capelli) [duplicate]

Let $F$ be a field of arbitrary characteristic, $a\in F$, and $p$ a prime number. Show that $$f(X)=X^p-a$$ is irreducible in $F[X]$ if it has no root in $F$. This answer to a related question ...
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### Check the proof of:$x^2+1$ is irreducible over $\Bbb Q$

Prove: $x^2+1$ is irreducible over $\mathbb{Q}$ Proof: Since $x^2+1=(x-i)(x+i)$, so $x^2+1$ is irreducible over $\mathbb{Q}$. Is it right?
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### Irreducibility of $f(x)=x^4+3x^3-9x^2+7x+27$

Question at hand is: Is $x^4+3x^3-9x^2+7x+27$ irreducible in $\Bbb Q$ and/or $\Bbb Z$. This is for an exam, reasoning is trivial, but no calculators in hand. Clearly, if there is a rational root, ...
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### Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$
How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don't exactly know how that applies. Thanks for any input.