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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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Is $x^p+a$ always irreducible in $\mathbb{F}_p[X]$?

As in the title. My first thought would be something using Fermat's little theorem, but I'm not sure where to go from there.
user497174
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Find $(a,b)$ such that $x^2-bx-a$ is irreducible and $\begin{bmatrix}0&a\\1&b\end{bmatrix}$ has order $8$

I believe I might be having a bit of trouble with a question that asks me to find the values of $a$ and $b$ such that the polynomial $x^2-bx-a$ is irreducible and the matrix $\begin{bmatrix} 0 & a \\ 1 & b \end{bmatrix}$ has order $8$. This is not…
asdf
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Prove that $p(x)$ has no integer zeros.

Suppose $p(x)$ is a polynomial over $\mathbb Z$ such that there exists a positive integer $k$ for which none of the integers $p(1), p(2), \ldots , p(k)$ is divisible by $k$. Prove that $p(x)$ has no integer zeros. My attempts: I tried to prove it…
dssknj
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polynomial in 2 variable representing irreducible polynomial over one variable

Let $f(x,t)$ be a polynomial with coefficients from $\mathbb{Z}[x,t]$. I am wondering how to check for irreducibility of $f(x,t)$ over $\mathbb{Z}[t]$. Any help is kindly acknowledged. Thank you
Math123
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The splitting extention above Z_ (p)

F (x) is a polynomial on Z_ (p), and n is an irreducible polynomial. When a is the root of f (x), is the splitting field of f (x) on Z_ (p)? Using Frobunius automorphism we have found that a, a ^ p, ... a ^ (p ^ (n-1)) is also the root of f…
LeeHanWoong
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Symmetry between roots of an irreducible polynomial in $\Bbb{Q}[x]$.

Let $f(x)$ be an irreducible polynomial in $\Bbb{Q}[x]$. I was trying to study these polynomials, and I notice a peculiar symmetry amongst the roots, although I'm not always sure what kind of symmetry the roots have. For instance, the polynomial…
user67803
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Irreducible polynomial of degree 3

$$P(X)= 21X^3 -3X^2+2X+9$$ To check whether it is irreducible or not in $Q[X]$. Since it's degree $3$ if it has a rational root then it is reducible as one of them would be linear factor; but how to show whether a polynomial of degree three has…
Tutankhamun
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Prove that the polynomials are irreducible on F5[x,y]

$ y^3 − (x^2)y + x(y^2) − 3x$ in $F5[x, y].$ Is it right to go through all the $(x,y)$ from $0,1,2$ and $4$ but when $x=0,y=0$ and $x=0,y=4$ and some other situations that the polynomial can be $0,$ does it mean that polynomial is reducible
vivi.wang
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Prove that the following polynomials are irreducible:

Prove that the following polynomials are irreducible: $x^6 − y(y − 1)x^4 + (y − 1)x^3 − (y^2 − 2y + 1)x + y − 1 \in R[x, y]$ i factor it into $x^6-y(y-1)x^4+(y-1)y^3 -((y-1)^2)x+(y-1) $ I don't know what to do next, is there anything to do $(y-1)$?
vivi.wang
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polynomial, reed solomon

Please advise if somebody knows: The polynomial x^2+x+2 ∈ F3^2is irreducible and i am looking for the generator polynomial, the length and dimension. What i did its: Find the vector-exponent equivalence for each element 0->…
Irene
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Show that if $a_{0}+a_{1}x+...+a_{n}x^{n}$ is irreducible in K[X] then $a_{n}+a_{n-1}x+...+a_{0}x^{n}$ is also irreducible.

Let K be a field and let $$ f(x) = a_{0}+a_{1}x+...+a_{n}x^{n} $$ is irreducible in K[X] then $$ a_{n}+a_{n-1}x+...+a_{0}x^{n} $$ is also irreducible.
user391148
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Is $x^3+y^2x^2+3yx-y$ irreducible in $\mathbb{Q}[y][x]$?

As the title says is $$x^3+y^2x^2+3yx-y$$ irreducible in $\mathbb{Q}[y][x]$ ? Apparently the answer is that it is indeed irreducible. My attempt: My first thoughts are what is $\mathbb{Q}[y][x]$, to which I reckon it's something like…
pshmath0
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Factorization of $f(x)=x^4-7x^3+10x^2+5x-10$

So the question is to show that $f(x)=x^4-7x^3+10x^2+5x-10$ factors over $\mathbb{Q}$ with a linear factor and a third degree polynomial that is irreducible. My solution: For $f(2)=0$ we can reduce $f(x)$ to $f(x)=(x-2)(x^3-5x^2+5)$, by Eisensteins…
ProShitposter
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Irreducible polynomials representation

Can anyone prove it? I should prove the existence and unique: Let p be an irreducible polynomial of k[x] of degree m. Prove that every element of k[x]/(p) can be represented uniquely by an expression of the form a1x^(m−1) + a2x^(m−2) + · · · +…
Jahangirova
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Show irreducible in k(x,y)[z]

k is a field and $n\geq 1$. Show that $z^n+y^3+x^2 \in k(x,y)[z]$ is irreducible. Can someone give hints? I am not sure how to apply Eisenstein's criterion to show irreducibility. Thank you
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