Linked Questions

6
votes
4answers
12k views

How to find all irreducible polynomials in Z2 with degree 5? [duplicate]

I am totally lost on how to do this one. I am supposed to accomplish the following: Find all irreducible polynomials in $\mathbb{Z}_2[x]$ with degree $5$. I may use the fact that x, $x+1$ and $x^2+x+...
1
vote
2answers
1k views

Determine all monic irreducible polynomials of degree $4$ in $\mathbb{Z_2[x]}$ [duplicate]

Determine all monic irreducible polynomials of degree $4$ in $\mathbb{Z_2[x]}$ Well these polynomials will be of the form - $a_0 + a_1x + a_2x^2 + a_3x^3 + x^4$ So we have four coefficients that ...
0
votes
1answer
111 views

Show that polynomial is irreducible [duplicate]

I am trying to prove that the polynomial $P = X^5 + X^2 + 1 ∈ F_2 [X]$ is irreducible. What I did: I showed that $X^2+X+1∈F_2[X]$ is the only irreducible polynomial of degree 2. Is there a way to ...
0
votes
0answers
37 views

How to find all monic irreducible polynomials of degree 5 in $F_{2}[x]$? [duplicate]

I already know how to get the number of monic irreducible polynomials of degree 5 in $F_{2}[x]$. that is $$I_{n}=(1/n)\prod_{d,d|n}\mathcal{U} (x)q^{n/d}$$ But I need to find all monic irreducible ...
0
votes
0answers
24 views

Finding all degree 4 irreducible polynomials over $\mathbb{Z_2}$ [duplicate]

So I'm looking for all the irreducible degree four polynomials over $\mathbb{Z_2}$. Now, there are only 16 total degree four polynomials over $\mathbb Z_2$, and so I could go through and check each ...
5
votes
3answers
134 views

Polynomial factorization into irreducibles over $\mathbb{Q}[x]$

I need to find irreducible factors of $f(x)=x^4+3x^3+2x^2+1$ in $\mathbb{Q}[x]$ and explicitely prove that these factors are indeed irreducible. I believe we can't reduce $f(x)$ any further but I ...
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 ...
3
votes
2answers
397 views

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$.

Show that $x^5-x^2+1$ is irreducible in $\mathbb{Q}[x]$. I tried use the Eisenstein Criterion (with a change variable) but I have not succeeded. Thanks for your help.
2
votes
1answer
613 views

Exhaustively generating irreducible polynomials over a Galois field

I'm working on generating de Bruijn sequences using a non-binary LFSR (as described in [1]). One problem I'm running into is finding all irreducible polynomials which can be then used to parametrise ...
3
votes
2answers
114 views

Factoring $x^{7} - 1$ into irreducibles over $\mathbb{F}_{2}[x]$

I know this breaks down into $(x - 1)(x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x +1)$, so the task is to show whether the second factor is irreducible over $\mathbb{F}_{2}[x]$. It's quick check that ...
1
vote
3answers
143 views

Polynomial decomposition into irreducible factors

Decompose $x^5 + x + 1$ into irreducible factors in $\mathbb{Z}_2[x]$. I would like to know how to reason and how to proceed. I am a beginner in this field of mathematics, and I am trying to ...
2
votes
1answer
143 views

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
1
vote
1answer
98 views

I have to show this polynomial is irreducible. [closed]

Suppose that $p(x)=x^9+x^8+x^4+x^2+1 \in \mathbb{Z}_2[x]$. I have to show this polynomial is irreducible.
0
votes
1answer
266 views

Factor a polynomial over a finite field

I have a polynomial: $x^6 + x^4 + x^3 + x + 1$, and I need to factor it over $\mathbb{F}_2$, $\mathbb{F}_4$ and $\mathbb{F}_{16}$. I need to do it manually. I wasn't even able to factor it over $\...
1
vote
2answers
114 views

Factor $x^8-x$ in $\Bbb Z[x]$ and in $\Bbb Z_2[x]$

Factor $x^8-x$ in $\Bbb Z[x]$ and in $\Bbb Z_2[x]$ Here what I get is $x^8-x=x(x^7-1)=x(x-1)(1+x+x^2+\cdots+x^6)$ now what next? Help in both the cases in $\Bbb Z[x]$ and in $\Bbb Z_2[x]$ Edit: I ...

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