All Questions
172
questions
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Number of irreducible polynomials of degree at most n over a finite field
We know that the number $N(n,q)$ of irreducible polynomials of degree $n$
over the finite field $\mathbb{F}_q$
is given by Gauss’s formula
$$N(n,q)=\frac{q-1}{n} \sum_{d\mid n}\mu(n/d)q^d.$$
The number ...
0
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2
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82
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Is this $f(x) = x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ irreducible in GF(5)?
Perhaps one can somehow apply Eisenstein's sign here by considering $f(x+1)$, but by default it is formulated for the expansion over $\mathbb{Q}$ of a polynomial from $\mathbb{Z}[x]$. Here we have $GF(...
- 15
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54
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When is $f = X^4 -1 \in \mathbb{F}_p[X], p $prime, irreducible and/or seperable? [duplicate]
I'm having some trouble figuring out a solution to this. I understand that $f$ is separable, iff all its roots are distinct, however I'm completely clueless about how to investigate that criterion......
- 17
6
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2
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151
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Irreducible polynomial in $\Bbb{Z}_2[x]$
Suppose $2k + 1 \equiv 3 \mod 4$ in $\Bbb{Z}_{\geq 1}$.
Is the polynomial: $p_k(x) = x^{2k + 1} + x^{2k - 1} \dots + x + 1$ irreducible in $\Bbb{Z}_2[x]$?
I do not know whether it is true or not...
(...
- 515
1
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1
answer
68
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Irreducible polynomials in $\mathbb F_q[T]$
Let $q$ be a power of a prime $p$. Is there an infinite set $S$ of $\mathbb N$ such that for every $l\in S$, the polynomial $T^{q^l}-T-1$ is irreducible in $\mathbb F_q[T]$.
It looks like Artin-...
- 1,157
0
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1
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93
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$
How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$?
I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
- 2,636
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0
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32
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A faster way to check irreducibility of quartics in finite fields
A quadratic or a cubic can be shown to be irreducible in $\mathbb{F}_p$ by showing that none of $0, \dots,p-1$ are roots (or more generally all the elements of $\mathbb{F}_{p^n}$). This does not work ...
- 3,940
3
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1
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153
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Characterization of irreducible polynomials over finite fields - alternative proof?
By accident I have found the following characterization of irreducible polynomials over $\mathbb{F}_p$.
Lemma. Let $g \in \mathbb{F}_p[T]$ be a monic polynomial of degree $m \geq 1$. Then, $g$ is ...
- 165k
1
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1
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69
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$p$ a prime satisfying $p \equiv 3 \mod 4 $. Then, the quotient field $ F_p [x] / (x^2 + 1)$ contains $\bar{x}$ that is a square root of -1
I know that $x^2 + 1$ is irreducible in $F_p[x]$ if and only if $-1$ is not a square in $F_p$. Otherwise, $x^2 + 1$ could be factored out.
$-1$ not being a quadratic residue in $F_p$ is equivalent to ...
- 155
8
votes
2
answers
267
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How to factor a polynomial quickly in $\mathbb{F}_5[x]$
I was doing an exercise in Brzezinski's Galois Theory Through Exercises and needed to factor the polynomial
$x^6+5x^2+x+1=x^6+x+1$ in $\mathbb{F}_5[x]$. Is there a quick way to do this?
I can see it ...
- 11k
0
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1
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286
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XOR-Product Modulo Prime
Every natural number seems to map to a polynomial in binary field GF(2). For example, $11 = 1011_2 \mapsto x^3 + x + 1$, and $x^3 + x + 1 \mid_{x=2}$ gives 11. How naturally can I go between natural ...
- 10.9k
4
votes
2
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228
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Help with a finite field exercise. How to find the minimal polynomial of a given root in a given field.
I need a help with this exercise.
(i) Find a primitive root $\beta$ of $\mathbb{F}_2[x]/(x^4+x^3+x^2+x+1)$.
(ii) Find the minimal polynomial $q(x)$ in $\mathbb{F}_2[x]$ of $\beta$.
(iii) Show that $\...
- 391
0
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0
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37
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Polynomial of degree 3 with coefficients over $\mathbb{F}_3$ has always a root in GF(27) [duplicate]
Prove or give a counterexample:
Every polynomial of degree 3 having coefficients over $\mathbb{F}_3$ has always a root in $\mathbb{F}_{27}$.
I noticed that $\mathbb{F}_{27} = \mathbb{F}_{3}[x]/(f)$, ...
- 87
2
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0
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161
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Splitting field of $x^8-1$ over $\mathbb{F}_2 ,\mathbb{F}_3,\mathbb{F}_{16}$
Find the splitting field of $f(x)=x^8-1$ over $\mathbb{F}_2$, $\mathbb{F}_3$ and $\mathbb{F}_{16}$.
I tried this: We claim that the field with $q=p^m$ elements is unique. A field with $q$ elements is ...
0
votes
1
answer
405
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Irreducible polynomial in integers modulo p
I am a completing a past paper question and I am undecided on what method to use here. The question is:
For what $a$ is $f(x)=x^3+x+a\in\mathbb{Z}_{7}[x]$ irreducible? My ideas are:
(1) Check each $a\...
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