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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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Finding minimal polynomial of $\sqrt3 + i \sqrt 7$ over $\Bbb Q$

$x=\sqrt 3 + i \sqrt 7$ I have come to the following polynomial: $x^4 + 8x^2+100=0$. By Eisenstein criterion, this pol is not irreducible in $\Bbb Q[x]$, but I don't know how to factor it further? Will be great if I can get some hints.
GalaxyY
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Irreducibility of $x_1^2+x_2^2+...+x_n^2$ in $\mathbb{C}[x_1,...,x_n]$

I wanted to know whether the polynomial function $x_1^2+...+x_n^2 \in \mathbb{C}[x_1,...,x_n]$ is irreducible in $\mathbb{C}[x_1,...,x_n]$ for $3 \leq n$ ?
Koalalover
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Properties of Integer Polynomials taking prime values infinitely often.

Let $f(x) \in \mathbb{Z}[X]$ such that the sequence $(f(n))_{n\in\mathbb{N}}$ contains prime numbers infinitely often. Then $f(x)$ is irreducible. The leading coefficient of $f$ is positive. The set $\{f(n):n \in \mathbb{Z}\}$ has no common…
Saikat
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Ist $3x^3 + 3x$ irreduzibel in $Z[x]$? In $Q[x]$?

Attempt: Let $(R,+,*)$ be an integral domain. A non-zero, non-unit polynomial $f(x)\in R[x]$ is called irreducible polynomial over $R$ if $f(x) = g(x)h(x)$, $g(x)\in R[x]$ and $h(x)\in R[x]$ then either $g(x)$ is unit or $h(x)$ is unit. So we can…
Vek
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Techniques of proving a polynomial is irreducible in rational numbers

I have two questions. I want to prove $x^3-6x^2+12x+7$ is irreducible in rational numbers. My attempt is to use Gauss's lemma, that a primitive polynomial is irreducible in integers iff it is irreducible in rationals. And in integers, it is enough…
Nonenicht
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Use of the Eisenstein Criterion

I am looking at the Eisenstein criterion and this works for unique factorization domain. Now I am asking myself if $\Bbb Q$ is a unique factorization domain? I would yes because it is a field. Now why we use often the criterion to $\Bbb Z$ and…
Frederick Manfred
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Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^3+6x^2+8x+4$

Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^3+6x^2+8x+4$ Dear Community, it took me some hours to think about it and i cant come to a good solution, i would be so happy if someone can show me how to do this…
Vek
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How do we check that a polynomial is irreducible in $\mathbb{C}[x,y]$

we know that $\mathbb{C}$ is an algebraically closed field so every polynomial $f$ in $\mathbb{C}[x]$ can be represented as $f=(x-a_{1})^{\alpha_{1}} ....(x-a_{n})^{\alpha_{n}} $ and the only irreducible polynomials are the constant polynomials…
Aster Phoenix
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Show $f(x) = 7x^4 + 103x^3 + 802x^2 + 6x + 75$ is irreducible in $\mathbb{Q}$

I want to show that $f(x) = 7x^4 + 103x^3 + 802x^2 + 6x + 75$ is irreducible in $\mathbb{Q}[x]$. My attempts are: Consider $f(x)$ reduced mod $2$. This gives $x^4 + x^3 + 1$ which is irreducible as it has no root and is not divisible by $x^2 + x +…
joan
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What are the possible degrees for $\Bbb C[x]$

If $f(x) \in \Bbb C[x]$ is a non constant irreducible polynomial, then what are the possible degrees of $f(x)$? What if $f(x)$ is in $\Bbb R[x]$?
nancy
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$\mathbb{R} [ X ] / ( X^ 2 + 2 )\cong \mathbb{C}$?

I am puzzled since in my eyes I cannot prove that there can be a function which is homomorphic regarding multiplication. Do you know any functions that could be? I was thinking that the quotient field has the format: < ax+b | x^2 = -2> Am I…
Petros Hatzivasiliou
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Show that $x^n-x-1$ is irreducible over $\mathbb{Z}$

Assume it is reducible, then $g(x)h(x)=x^n-x-1$ Note $g(0),h(0),g(1),h(1)=\pm1$ Then I am stuck.
Kai
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Irreducible polynomials?

Is the polynomial $x^n+2x^{n-1}+\dots+nx+n+1$ irreducible in $\mathbb Z[x]$?
MarkoR
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regarding Integer solutions

Let $f(x)=x^7-105x+12.$ Show that $f(m)$ is not prime for any integer $m$.
Ravindra
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Decide if f(X) is irreducible in the following rings

Let $$f(X) = 78X^3 + 174X^2 − 116 ∈ Z[X]$$ My question is to decide if $f(X)$ is irreducible in $Z[X], Q[X] and R[X]$ I have tried finding a prime number 29, and to fulfil the Eisenstein's Irreducibility Criterion 1) $29$ is a common factor of…
Thomas
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