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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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$f(x)=x^m+1$ is irreducible in $\mathbb{Q}[x]$ if only if $m=2^n$.

I have a question Prove that : $f(x)=x^m+1$ is irreducible in $\mathbb{Q}[x]$ if only if $m=2^n$ Thanks for your helps!
user41499
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If p(x) $\in F[x]$ is of degree 3, and $p(x) = a_0 + a_1*x + a_2*x^2 + a_3*x^3$.

If p(x) $\in F[x]$ is of degree 3, and $p(x) = a_0 + a_1*x + a_2*x^2 + a_3*x^3$, show that p(x) is irreducible oer F if there is no element $r \in F$ such that $a_0 + a_1*r + a_2*r^2 + a_3*r^3$. So far I've said that p(x) is not irreducible iff it…
user167000
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Wikipedia incorrect about irreducibile polynomials and roots

Wikipedia states : If an univariate polynomial $p$ has a root (in some field extension) which is also a root of an irreducible polynomial $q$, then $p$ is a multiple of $q$, and thus all roots of $q$ are roots of $p$; this is Abel's …
Cris
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Checking if a monic polynomial can be decomposed into linear factors

I have questions about how to determine if a polynomial can be decomposed into linear factors. If it is not solvable over radicals by Galois Theory, then I am done. But do I have to resort to Galois Theory? Let the polynomial be: $$f(x) = x^5 + a…
user152331
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Factorization of Polynomials. Irreducible polynomial (basic question)

One of the first examples says that: Let $f(x) = 2x^2 +4$. $f(x)$ is reducible over $\mathbb{Z}$ $f(x)$ is irreducible over $\mathbb{Q}$ $f(x)$ is irreducible over $\mathbb{R}$ $f(x)$ is reducible over $\mathbb{C}$ Why? For the first one, I see…
José D.
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$1+x^p+x^{2p}+\dotsb+x^{p(p-1)}$ irreducible

Let $p$ be a prime number. Is $$f(x)=1+x^p+x^{2p}+\dotsb+x^{p(p-1)}$$ an irreducible polynomial over $\Bbb Z$? Can we use Eisenstein's criterion? $f(x+1)=1+(x+1)^p+(x+1)^{2p}+\dotsb+(x+1)^{p(p-1)}$ I am stuck. Thanks a lot!
ziang chen
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Check if polynomial is minimal over $\mathbb{Q}$

I want to determine the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb{Q}$. I calculated the polynomial $x^4-10x^2+1$. Now I used Eisensteins theorem to test if it is irreducible. It is not applicable in this form so I substituted $x$ with…
Dino Rossegger
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Show $5x^{n}-1 \in \mathbb{Q}[x]$ for $n \geq 1$ is irreducible.

$5x^{n}-1 \in \mathbb{Q}[x]$ for $n \geq 1$ is irreducible. I tried to prove it with Eisenstein's Criterion, but I did not know how to use it. I also used the fact that a polynomial f(x) over a field k is irreducible if the polynomial f(x+1) is…
The πzza man
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On irreducible polynom $\frac{X^p-1}{X-1}$

Let $m(X) := \frac{X^p-1}{X-1}$ and $n(X) := m(X^p)$. I have shown that $m$ is irr. over $\mathbb Q$. Now I want to show that this is true for $n(X)$, too. I know that $$ n(X+1)((X+1)^p-1)= (X+1)^{p^2}-1 $$ Working modulo $p$ I get that…
user42761
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Show $f(x)=x^2+1$ is irreducible over $\mathbb{Q} (i\sqrt{2})$

I am wanting to show $x^2+1$ is irreducible over $\mathbb{Q} (i\sqrt{2})$, however, I'm a little stuck. Here's my attempt: Suppose, for contradiction's sake, that $x^2+1$ is reducible, and factors as $(x-\alpha)(x-\beta)$. We can expand this as…
Moni145
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$x^2+y^n-1$ is absolutely irreducible

I have encountered this while going over the wikipedia page "irreducible polynomial". A polynomial that is irreducible over any field containing the coefficients is absolutely irreducible. By the fundamental theorem of algebra, a univariate…
テレビ スクリーン
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Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^5+x+1$

Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^5+x+1$ Attempt: We can write $x^5+x+1 = (x^2+x+1)(x^3-x^2+1)$ which could be a hint to be reducible, if we look clously we see that $(x^2+x+1),(x^3-x^2+1)$ are both…
Vek
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Irreducible polynomial for infinitely many values

I want to prove that there are infinitely many values of k such that the polynomial $x^{9}+12x^{5}-21x+k$ is irreducible. I sense that I have to use Eisenstein and the number 3 but I don't see exactly how. Any help would be appreciated.
Teplotaxl
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Prove polynomial is irreducible in $Z_p[x]$.

Give $f(x)=a_0+a_1x+\ldots+a_nx^n$ and prime $p$ that $p \nmid a_n$ and $GCD(a_1,a_2,\ldots,a_n)=1$. Which one in two clause below is correct? (1):"If $f(x)$ is irreducible in $\mathbb{Z}[x]$ then $f(x)$ is irreducible in $\mathbb{Z}_p[x]$. (2):"If…
Xiuwei Lee
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An Irreducible polynomial in $\mathbb{Q}[x]$

I want to prove that $x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Q}[x]$. I noticed that this polynomial can be rewritten as $x(x+1)(x^2+1)+1$. As seen, it has no proper divisor because of the constant term $1$. However, I need a rigorous proof. Can…
Hussein Eid
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