Linked Questions

3
votes
1answer
459 views

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 $\mathbb{Z}/p\mathbb{Z}$?
0
votes
0answers
87 views

How many irreducible factors of grade $6$ there is in $\mathbb{F}_{2}\left[ x\right]$? [duplicate]

How many irreducible factors of grade $6$ there is in the polynomial ring $\mathbb{F}_{2}\left[ x\right]$? I have solved this by using the fact that every irreducible polynomial of grad $i$ is a ...
0
votes
0answers
28 views

Is there a formula which would let me know how many irreducible polynomials there are to the power n, in $z_n$? [duplicate]

I found that $x^2+x+1$ is the only polynomial to the power 2 that is irreducible in $z_2$. Moreover I found that $x^3+x+1$ and $x^3+x^2+1$ are the only polynomials to the power 3 that are ...
2
votes
0answers
22 views

Number of irreducible polynomials of degree 4 on $\boldsymbol{Z}_3[x]$. [duplicate]

Find the number of irreducible polynomials of degree 4 in $\boldsymbol{Z}_3[x]$. Not really sure what to do here, I've tried listing them all out but this seems tedious, and all I need to find is the ...
0
votes
0answers
21 views

Find monic irreducible polynomial in p-field [duplicate]

Let $p$, $l$ be prime numbers, $n$ be positive integer. Find the number of irreducible element in $\mathbb {F_p}[x]$ with degree of $l^n$. I have got the Gauss's formula $\dfrac{1}{n} *\displaystyle\...
49
votes
2answers
17k views

Number of monic irreducible polynomials of prime degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$? Thanks!
14
votes
4answers
1k views

A field of order $32$

I was working on this problem from an old qual exam and here is the question. In particular this is not for homework. True or False: There are no fields of order 32. Justify your answer. Attempt: ...
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 $\...
9
votes
1answer
2k views

Lack of understanding of the proof of the existence of an irreducible polynomial of any degree $n \geq 2$ in $\mathbb{Z}_p[x]$

In an optional course called "Finite Geometries", we most recently constructed the fields $$K_{p,n}[x] := \{\alpha \in K_p[x]\,| \deg(\alpha) < n\},$$ where $K = \mathbb{Z}$, $p$ is prime and $n \...
2
votes
3answers
2k views

Constructing finite fields of order $8$ and $27$ or any non-prime

I want to construct a field with $8$ elements and a field with $27$ elements for an ungraded exercise. For $\bf 8$ elements: So we can't just have $\Bbb Z/8\Bbb Z$ since this is not even an integral ...
4
votes
2answers
427 views

Counting irreducible polynomials over finite fields [duplicate]

How many irreducible polynomials in $Z_2[x]$ of degree $3$? I have discussed this with my friend before and we found that $x^3 + x^2 + 1$ and $x^3+x+1$ are the two said polynomials which irreducible....
6
votes
1answer
429 views

Calculating the number of irreducible polynomials over a finite field

I am trying to find the number of irreducible polynomials of degree $n$ over $\mathbb{F}_p$. Here is what I have done: (1). Let $K=\mathbb{F}_{p^n}$. Let $M(n,p)$ the number of monic irreducible ...
2
votes
1answer
921 views

Patterns in $GF(2)$ Polynomial division.

I am testing Prime polynomials in $GF(2)$ and have noticed a pattern that I hope will help. There's a calculator here if you want to familiarise yourself with polynomials over $GF(2)$. I am testing ...
4
votes
2answers
194 views

Combinatorial solution to a recurrence problem

I found today this problem in some old blog post: Suppose that $a_n$ is a sequence of positive integers such that $$ \sum_{d |n} a_d =2^n$$ for every positive integer $n$. Prove that $ n | a_n$. It ...
3
votes
2answers
121 views

How do you discover an irreducible polynomial over a finite field that has a chosen degree?

I'm would like to find a few polynomials of degree 127 irreducible over $\mathbb{F}_2$ for use in constructing $GF(2^{127})$. This is because I'm thinking of trying to make my own version of AES-GCM ...

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