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Questions tagged [irreducible-polynomials]

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

An irreducible or prime polynomial is a polynomial that has no polynomial factors. The parallel with prime integers is appropriate. In the same way that a prime number cannot be divided by any other integer other than $\pm1$, an irreducible polynomial cannot be divided by any polynomial other than a constant non-zero polynomial.

Though it is usually clear from context, care should be taken in observing the domain over which a polynomial is irreducible. For instance, $x^2+4x+13$ is irreducible over $\mathbb{R}$ but not $\mathbb{C}$; likewise, $x^3 + 2 x^2 + x + 3$ is irreducible over $\mathbb{F}_{5}$ but factors as $(x-1)(x-2)^2$ over $\mathbb{F}_7$.

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Factor $x^{17}+1$ into a product of irreducibles

In title, factor $x^{17}+1$ into a product of irreducibles over $\mathbb{R}$. I know it factors as $$(x+1)(x^{16}-x^{15}+\dots+1)$$ but I have no real justification for why the second factor is irreducible besides "mathematica says it's true and I…
Joseph DeGaetani
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The polynomial $x^{2k} + 1 + (x+1)^{2k}$ is not divisible by $x^2 + x +1$. Find value of $k$ such that it belongs to natural numbers.

I did some progress by doing this- I thought if it would be a factor, then $x^2 +x +1 = 0 $ $x+1=-x^2$ Putting this in the expression, $$x^{4k}+x^{2k}+1$$ Then I tried to solve it further, but I am stuck here...
Aayush
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Is there something to say about the irreducibility of polynomials and their derivatives?

Is there a relation between the irreducibility of a polynomial and its derivative under certain conditions?
user157036
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Irreducible polynomial with three real zeros

Is there an irreducible (cubic) polynomial with rational coefficients with three real zeros? (When I speak of irreducibility I mean over rational numbers.) How about an irreducible polynomial of degree $n$ with rational coefficients with $n$ real…
Timotej
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Show that $Y^2 + X^2(X+1)^2$ is irreducible over $\mathbf R$

Show that $Y^2 + X^2(X+1)^2$ is irreducible over $\mathbf R$. Are there some general tricks for avoiding barbaric computations in general case?
user171326
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Prove that the following polynomials are irreducible or not.

I want to show that: 1) $X^4+1$ is irreducible The roots are the elements of $$\left\{\frac{\pm 1+i}{\sqrt 2},\frac{\pm 1-i}{\sqrt 2}\right\}$$ therefore it's not a product of a polynomial of degree 1 and a polynomial of degree 3. Is there an…
idm
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Is $x^{p}+\alpha$ irreducible over $\mathbb{F}_{p}(\alpha)$ for transcendent $\alpha$?

Let $\mathbb{F}_{p}(\alpha)/\mathbb{F}_{p}$ be a transcendent field extension and $f(x)=x^{p}+\alpha$. Then could it be that $f$ is irreducible over $\mathbb{F}_{p}(\alpha)$? And why or why not? I assume that it is.
Celsius
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irreducible polynomial in polynomial ring of 2 variables

So I've been asked to prove that $x^2+y^2$ is irreducible over $\mathbb R[x,y]$, the polynomial ring over $\mathbb R$ in two indeterminates $x$ and $y$. I don't want the solution. I just don't know if I am heading in the right direction. What I've…
user83610
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How can I determine which polynomials are irreducible?

For this problem, I was given three polynomials to determine which were irreducible: $x^4 + x + 1\in\mathbb Z/(2)[x]$ $1 + x + x^2 + ... + x^{41}\in\mathbb Z/(2)[x]$ $x^{42} + 42x + 4x^2 + 42\in\mathbb Q[x]$ I know that using Gauss' Lemma and…
user126731
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Proving irreducibility of polynomial

I need to show that $x^4+4x^3+6x^2+9x+11$ is irreducible in the integers. First, I tried to apply Eisenstein's irreducibility criterion by shifting $x$ to $x+\alpha$. However, I can't think of any shift to apply that would fit the criterion. Next, I…
Andrea
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Prove that if the coefficient of $ax^3+bx+c$ are odd then it is irreducible of $\mathbb{Q}$

Let $a,b,c$ be odd integers. Prove that $p(x)=ax^3+bx+c$ has no rational root.
Tulip
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Is $X^7 + X^4 + 1$ irreducible over $Q(X)$?

From the 3rd edition of the book "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan, we find the following exercise under chapter 4: Exercise 140: "Is the polynomial $X^7 + X^4 + 1 ∈ Q[X]$ irreducible?" At this…
Just_a_fool
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How to prove the irreducibility $x^m+x^n+x+1$?

Let $m>n>1$ be odd positive integers, show that $x^m+x^n+x+1$ is irreducible in $\mathbb{Z}[x]$. I guess the proposition is true in $\mathbb{Z}_3$, but have absolutely no idea about how to prove it.
OrthoPole
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Is the polynomial $x^5 - 5x^4 + 7x^3 + x^2 + x - 1$ irreducible in $\mathbb{Z} [x]$ or in $\mathbb{Q} [x]$?

Is the following polynomial irreducible in $\mathbb{Z}[x]$ or in $\mathbb{Q}[x]$? $x^5 - 5x^4 + 7x^3 + x^2 + x - 1$ If it's reducible, there should be a linear factor with degree $1, 2$ or $3.$ I try $\mathbb{Z}_2[x]$ and we get $x^5 +x^4 + x^3 +…
Vek
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Product of Roots

I recently tutored a neighbor's son who is a $11$th grade honor student at an academy which is not far from my residence. One of the questions of the extra credit that his teacher gave his class was: Question: Let $N > 2$ be the smallest integer…
DeepSea
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