# Tagged Questions

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

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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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### Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$? I tried to group it like $(4)x^2+(3y^2)x+(y^3+7)$. This is a polynomial with degree $2$ so I am thinking of applying quadratic formula... where ...
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### If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
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### Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
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### Irreducible polynomial - Artin-Schreier exercise

Can somebody help me with this exercise? Is $f(x)=50x^6 + 6x^5 -10x^4 + 15x^3 + 4x -7$ irreducible in $\mathbb Z[x]$? I know I have to project in $\mathbb Z_5$ and it becomes an artin ...
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### Number of irreducible polynomial over a field. [closed]

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements. Please anyone help me.
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### Can we continually factorize an expression like $x+y$?

I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ...
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### Irreducible polynomials over $\mathbb{Z}_2[x]$

Prove that the polynomial $1+x+..+x^m$ is irreducible over $\mathbb{Z}_2$ if and only if $m+1$ is a prime number and 2 is a primitive root in $\mathbb{Z}_{m+1}$ Is there any proof without using ...
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### Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
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### Irreducible polynomial of every degree over finite field

The existence of polynomials in title has been asked as a problem on MathStack many times; some answers were using existence of finite bigger fields, and some answers concern Mobius function with ...
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### Reducibility of $x^q -x -1$ in $\mathbb{F}_{q}$

I came across the following excercise and do not know how to go about this. Given the polynomial $x^q -x -1$ in $\mathbb{F}_{q}$. Consider $q=8$. Show this polynomial is reducible by considering an ...
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### For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible. What I tried: To show that $f$ is not a unit I did the following. Suppose that $f$ is a unit, then there exists ...
Let $K$ be a field, and consider a monic irreducible polynomial $f(x) \in K[x]$. Denote $d = \deg(f)$, and let $g(x) = f(x^2)$. Furthermore, let $\alpha \in \Omega^g_K$ (the splitting field of $g$ ...