Linked Questions

3
votes
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 $21.$ ...
0
votes
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.
11
votes
3answers
11k views

How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?

I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$. I am interested in counting how many such $...
6
votes
1answer
3k views

Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem?

Suppose $F_{p^k}$ is a finite field. If $F_{p^{nk}}$ is some extension field, then the primitive element theorem tells us that $F_{p^{nk}}=F_{p^k}(\alpha)$ for some $\alpha$, whose minimal polynomial ...
8
votes
2answers
5k views

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime. I'd like to start off by acknowledging that I know there are many posts relating to similar ...
8
votes
2answers
1k views

Counting Irreducible Polynomials

I'm investigating irreducible polynomials over finite fields at the moment, and I wanted to know if there is a formula for the number of irreducible polynomials of degree n over a fixed finite field $\...
8
votes
1answer
2k views

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 ...
2
votes
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 ...
17
votes
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 ...
6
votes
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 $R$?
2
votes
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 ...
1
vote
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 $\mathbb{F}_q$ ...
4
votes
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 ...
2
votes
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.
3
votes
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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