# Linked Questions

3answers
4k views

### Number of irreducible quadratic polynomials over a finite field [duplicate]

To find the number of irreducible polynomials of the form $x^{2} + ax+b$ over the field $F_{7}$ I manually checked all the possibilities and thus found the answer to be $21.$ ...
1answer
346 views

### Number of irreducible polynomials over $\mathbb Z_p$ [duplicate]

How many irreducible polynomials over $\mathbb Z_p$ of the form $x^2+ax+b$ are there? No idea.
3answers
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1answer
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### Monic Irreducible Polynomials over Finite Field

Let $F=\mathbb{F}_{q}$ be a finite field (so $q=p^k$ for some prime $p$ and positive integer $k$), and let $\varphi(d)$ denote the number of monic irreducible polynomials of degree $d$ in $F[X]$. I'm ...
2answers
445 views

### Proof existence of field extension of $\mathbb{F}_p$ containing the $r$-th primitive root of unity

I have to show the following: Let $p$ be a prime and $r \in \mathbb{N}$ with $\gcd(r,p)=1$. Prove the existence of a field extension $E$ of $\mathbb{F}_p$ which contains an $r$-th primitive root of ...
2answers
497 views

### How do mathematicians know what is known?

How do mathematicians know that what they are researching has not been already known for $200$ years? Obviously, if they are researching something that is cutting edge it is not a problem, but if one ...
1answer
392 views

### Irreducible polynomial in $\mathbb{F}_{p}[x]$

I'm studing for an exam and I am stuck on the following practice problem. Consider the the ring $R=\mathbb{F}_{p}[x]$. How many irreducible polynomials of degree 4 exist in $R$?
3answers
211 views

### To Factorize $x^{27}-x$ over $\mathbb F_3$.

Problem 7.5 in Chapter 15 of Artin's Algebra asks to factorize $x^{27}-x$ over $\mathbb F_3$. Here is what I have done. $x^{27}-x=x(x^{26}-1)= x(x^{13}-1)(x^{13}+1)$. In am having trouble ...
2answers
633 views

### Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field $\mathbb{F}_q$ ...
4answers
79 views

### Showing existence of irreducible polynomial of degree 3 in $\mathbb{F}_p$

I'am trying to show that for every p$\in \mathbb{N}$ where p is prime, there is an irreducible polynomial of degree 3 in $\mathbb{F}_p$. I've found too general answers for that question, but I want ...
2answers
123 views

### 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.
1answer
229 views

### polynomials over finite field with irreducible factors of odd degrees

It is well-known that the number of monic $n$-degree polynomials over a finite field of size $q$ is $q^n$. How many such degree-$n$ polynomials can be completely factored into only irreducible ...

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