All Questions
72
questions
2
votes
1
answer
73
views
Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?
I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
- 474
3
votes
0
answers
109
views
Roots of a irreducible polynomial are linearly independent over a finite field.
Question. When does a irreducible polynomial contains linearly independent roots over a finite field?
Motivation. For a finite cyclic Galois extension $E/F$, if $\alpha\in E$ generates a normal basis, ...
- 2,503
0
votes
1
answer
201
views
Splitting field of $x^3 +x +1$ over $\mathbb F_{11}$
This is a HW problem for an algebra course.
Determine the splitting field of $f(x)=x^3+x+1$ over $\mathbb F_{11}$.
I tried to use the answers from this question and this question to help me, but want ...
- 2,454
1
vote
3
answers
99
views
Number of roots of $f=x^5-x-1$ in $\mathbb{F}_4$
According to David Cox’s ‘Galois Theory’ (Proposition 11.1.5.),
If $f \in \mathbb{F}_p[x]$ is nonconstant and $n \geq 1$, then the
number of roots of $f$ in $\mathbb{F}_{p^n}$ is the degree of the
...
- 1,788
2
votes
1
answer
78
views
Prove that element is a square (follow up).
I asked the following question and got some awesome answers:
Suppose that $x^5 + ax + b \in \mathbb{F}_p[x]$ is irreducible over $\mathbb{F}_p$. Is it true that $25b^4 + 16a^5$ is a square in $\...
- 169
2
votes
2
answers
850
views
Generating irreducible polynomials for binary numbers
From a paper, there is a discussion about generating irreducible polynomials for a certain degree as can be seen below.
The reference is wolfram, but it is not clear how those polynomials are ...
- 223
0
votes
0
answers
145
views
Evaluation of a polynomial over the finite field $\mathrm{GF}\left(2^{8}\right).$
I am trying to make a program that, among other things, considers a polynomial $p$ whose coefficients are elements of $\mathrm{GF}\left(2^{8}\right)$ and shows the user the graph of that polynomial. ...
- 152
2
votes
1
answer
311
views
Modulo calculation on a polynomial, in NASA tutorial on Reed-Solomon codes
I am reading Geisel's tutorial$^{\color{red}{\star}}$ on Reed-Solomon codes, in which a Galois Field is developed. The elements of the field are generated as consecutive powers of $X$, modulo an ...
- 143
0
votes
1
answer
200
views
Irreducibility over finite fields and the integers
It is widely known that if a univariate polynomial $f(x)= \sum_{i} a_ix^i$ is irreducible over a finite field $\mathbb{F}_p$ for some prime $p$ not dividing the leading coefficient of $f(x)$ then the ...
- 1
2
votes
1
answer
217
views
Polynomial over a finite field is a composition of a separable polynomial and a $x^{p^e}$.
In the proof of Corollary 3.2 (in the paper http://www.math.iitb.ac.in/~srg/preprints/CarlitzWan.pdf)
asserts the following:
Let $\mathbb{F}_q$ be the finite field of $q$ elements and $p=char(\mathbb{...
- 105
1
vote
1
answer
178
views
Decompose in irreducible factors over $\mathbb{Z}_3[X]$
I've been solving some problems from my Galois Theory course, but I can't solve this one and don't find help in my course notes. It says:
Decompose in product of irreducible factors over $\mathbb{Z}...
- 3,564
0
votes
0
answers
46
views
Irreducible polynomial in $\mathbb{Z}_{17}$ [duplicate]
Is the polynomial $x^{10}+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$ irreducible in $\mathbb{Z}_{17}$? I don't know how to solve it.
Thank you for any help!
- 449
1
vote
0
answers
69
views
Consequence of Chebotarev's density theorem
I'm studying Chatzidakis Notes about pseudo-finite fields. During a proof she states the following (by $\mathbb{F}_p$ I mean the finite field with $p$ elements for a prime $p$):
Let $f_1(x),\ldots,f_m(...
- 47
0
votes
1
answer
73
views
$x^n - 1$ factors in finite field
I am having some difficulty with some Galois problems and I was wondering if it is always true that
$x^n-1$ is a factor of $x^{nm}-1$ in a finite field,
and if not what conditions need to be imposed.
...
- 45
3
votes
0
answers
113
views
Irreducible factorization over $\mathbb{Q}$
I have implemented Berlekamp-Hensel algorithm that provide to factor (into terms of irreducible polynomials) primitive and square-free polynomial over $\mathbb{Z}$. At first, this algorithm obtains ...
- 41