Finite fields are structures arising in abstract algebra. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.
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41 views
Chaos in finite field
Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map
$x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $
where $\mathcal{P}$ - ...
1
vote
1answer
25 views
If $F$ is a finite field of size $q=p^n$ and $b\in K$ is algebraic over $F$ then $b^{q^{m}} = b$ for some $m > 0$
I want to apply a similar type of argument to show that $\alpha^{q} = \alpha$ for all $\alpha\in F$. When we know that the characteristic of $F$ is $p$ and $|F| = q = p^{n}$. But I dont know how to ...
2
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1answer
47 views
Basic concepts in finite fields
I need some help with clearing up some some basic concepts in finite fields.
I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to ...
4
votes
2answers
39 views
Linear polynomials of finite fields
I have a final tomorrow, and I was looking over some exercises in my textbook. However, I can't seem to work this problem out.
Let $F$ be a field of $p^n$ elements and let $\alpha \in F^*$, where ...
3
votes
2answers
38 views
Irreducible polynomials have distinct roots?
I know that irreducible polynomials over fields of zero characteristic have distinct roots in its splitting field.
Theorem 7.3 page 27
seems to show that irreducible polynomials over $\Bbb F_p$ have ...
4
votes
4answers
65 views
Irreducible Polynomial in $\mathbb F_{256}$.
Let $\mathbb F_{256}$ be the finite field with $2^8 = 256$ elements. Consider the polynomial over this field
$$
x^2 + x + 1.
$$
I wanted to know if it is irreducible, so I calculated it for all ...
2
votes
0answers
45 views
Irreducible polynomial roots and representations for Galois field elements in normal basis
I am trying to understand some topics in the literature and came across the following problem. Say I have a field $GF(2^4)$ defined by the irreducible polynomial $r(z) = z^4 + z + 1$, and I want to ...
0
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0answers
27 views
Elements of the form $aX^2 + bY^2$ in a finite field.
For cardinality reasons, we know that every element in a finite field $F$ is a sum of two squares. If I fix $a,b\in F$ with $a,b\neq 0$, can every element in $F$ be written in the form $aX^2 + bY^2$ ...
4
votes
1answer
25 views
equality between the index between field with $p^{n}$ elements and $ \mathbb{F}_{p}$ and n?
can someone explain this?
$ \left[\mathbb{F}_{p^{n}}:\mathbb{F}_{p}\right]=n $
11
votes
0answers
88 views
Combinatorics in finite vector space
Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$.
Let $k$, $a$ and $b$ be non-negative integers.
Determine the number of subspaces $K$ of $V$ ...
12
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5answers
696 views
Do we really need polynomials (In contrast to polynomial functions)?
In the following I'm going to call
a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication)
that has the form ...
3
votes
2answers
50 views
Geometric solution to classic committee problem
Most people know the classic committe style problems.
I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works.
I was hoping someone ...
0
votes
1answer
38 views
Are all of the irreducible representations of (any) symmetric group over $\mathbb{C}$ also irreducible over a finite splitting field.
This is probably an incredibly stupid question, but I'm a novice to representation theory and finite field theory, as I've just been introduced to these concepts, so all I really need is confirmation ...
0
votes
1answer
34 views
Finding squares in finite fields
I need the definition of finding squares in finite fields and also the number of squares in a finite field. How can we find squares in $\Bbb F_5$ and $\Bbb F_7$? (Here $\Bbb F_5$ and $\Bbb F_7$ ...
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votes
0answers
40 views
A question about hyperplanes in affine geometries [closed]
List all hyperplanes in
$\operatorname{AG}_3(2)$
$\operatorname{AG}_4(2)$
What is the main idea while listing?
Can you explain please?
3
votes
2answers
41 views
Order of the unit group of a finite field F if for all two subgroups of F one is contained in the other.
Let $F$ be a finite field. Prove that the following are equivalent:
i) $A \subset B$ or $B \subset A$ for each two subgroups $A,B$ of $F^*$.
ii) $\#F^*$ equals 2, 9, a Fermat-prime or $\#F^* -1$ ...
1
vote
1answer
54 views
Solve equations in a field with characteristic p.
Let $p$ be a prime and $K$ a field with characteristic $p$. How to solve the equation $x^2+2=0$ in the field $K$? Here $x, 2, 0 \in K$. Is it equivalent to solve the equation $x^2+2 = 0 \pmod p$? What ...
8
votes
3answers
188 views
Solve $x^3+x \equiv 1 \pmod p$
Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions.
We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$
We can prove that $(1)$ cannot ...
5
votes
2answers
104 views
Seeing that $\Bbb F_2[x]/(x^2+x+1)$ is a field
I have some basic question with polynomials appreciate if someone could explain me this.
Following is additional and multiplication tables and it is say that this is a field. Have no idea why say ...
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0answers
55 views
Full Rank Matrix with a specific construction
Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$
$$Z=\begin{bmatrix}
w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
6
votes
3answers
77 views
Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies
Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime.
Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
3
votes
1answer
76 views
Irreducible polynomial over $\mathbb Z/7\mathbb Z$
Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
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1answer
39 views
Show that for a finite field $F$ and finite group $G$, $F$ is a splitting field for $FG$
Let $F=\mathbb{F}_q$ be the field of $q$ elements and $G$ be a finite group. I'm trying to show that for an irreducible $FG$-module $V$, we have $\mathrm{End}_{FG}(V)=F \cdot 1$, i.e. that $F$ is a ...
1
vote
1answer
44 views
Finding quadratic residues in a finite field by using a primitive element
Let $1+2x$ be a primitive element of the field $\mathbb F_9$ obtained via the irreducible polynomial
$$x^2 + 1$$
over the base field $\mathbb F_3$.
i) Make a list of the elements of $\mathbb F_9$ ...
3
votes
5answers
57 views
On any finite field, adding the identity element a finite amount of times will result to the neutral element
Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$).
I started with saying that in every field there's an element $a \in F$ so that $a + ...
3
votes
1answer
44 views
Lagrange interpolation of Galois field functions
I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
4
votes
3answers
76 views
Order of matrices in $SL_2({\mathbb{F}_q})$
Could you tell me how to prove that in $SL_2({\mathbb{F}_q})$ the only element of even order is $-I$ ($ \ I$ - identity matrix)?
I would really appreciate a thorough explanation, because I cannot ...
2
votes
4answers
81 views
Finite fields and primitive elements
Let $\mathbb F_9$ be a finite field of size $9$ obtained via the irreducible polynomial $x^2 + 1$ over the base field $\mathbb F_3$.
How can you find a primitive element?
Make a list of the ...
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0answers
37 views
Let $p$ be a prime. $f(x) = 1 + x + x^2 + … + x^{p-1}$. Show that for any field F the irreducible factors of f(x) in F[x] all have the same degree
Let $p$ be a prime and set $f(x) = 1 + x + x^2 + ... + x^{p-1}$.
Show that for any field $F$ the irreducible factors of $f(x)$ in $F[x]$ all have the
same degree.
If $\operatorname{char}F=0$ , then ...
2
votes
1answer
45 views
Question about terminology, finite fields
My English is not very good, and that's why I would really appreciate it if you could explain to me what the phrase : these elements are under the same domain under
$F$ and $\alpha$ means in this ...
1
vote
1answer
38 views
Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$
I have this exercise :
$q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
2
votes
1answer
58 views
Counting roots of a multivariate polynomial over a finite field
How many roots can there be of a polynomial $f \in K[x_1, x_2, \dots , x_n]$ where $K$ is a finite field and the maximum exponent of $x_i$ in any term is $m$ for all $i$, assuming not all coefficients ...
1
vote
1answer
60 views
Rational and irrational fractions over finite fileds [duplicate]
I've been told that over the field $\mathbb{F}_7$ the square root of $2$ is actually $3$.
How come? Why does it happen?
1
vote
2answers
87 views
Roots of polynomials over finite fields
I've been trying to find the decomposition of $x^2-2$ to irreducible polynomials over $\mathbf{F}_5$ and $\mathbf{F}_7$.
I know that for some $a$ in $\mathbf{F}_5$ (for example), $x-a$ divides $x^2-2$ ...
0
votes
1answer
39 views
Polynomials decomposition into irreduceables
I've been trying to find the composition to irreduceables of the following polynomials with no much success:
X^2 +1 over the field F7
and X^2-2 over the field F5
Is there any method/algorithms I ...
1
vote
2answers
54 views
Maps compatible with the Frobenius
Let $F$ be a field. Fix a separable closure $F^{sep}$. Consider the monoid whose elements are maps of sets $F^{sep} \to F^{sep}$ which are equivariant with respect to the Galois action. These maps ...
4
votes
0answers
77 views
Minimal polynomial of a finite purely inseparable field extension
Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
4
votes
0answers
96 views
Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$
I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$.
Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines.
There are $q+1$ ...
2
votes
2answers
132 views
The linearity of the extended code
If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is:
If the code $C$ is linear, can we prove that the extended code $C'$ is linear too?
...
7
votes
5answers
186 views
Are there any other constructions of a finite field with characteristic $p$ except $\Bbb Z_p$?
I mean, $\Bbb Z_p$ is an instance of $\Bbb F_p$, I wonder if there are other ways to construct a field with characteristic $p$?
Thanks a lot!
2
votes
3answers
69 views
Show that A is a field which has $9$ elements
For $A=\left\{\left( {\begin{array}{*{20}{c}}
a&b\\
{ - b}&a
\end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
0
votes
1answer
34 views
Lin Alg- Dual Spaces
Let $(V^*)^*=V^{**}$. Define $S:V\to V^{**}$ by $s(v)(\alpha)=\alpha(v)$ for all $v\in V$ and $\alpha\in V^*$.
I need to show that $s(v)\in V^{**}$. And show the S is a linear transformation. ($V$ is ...
2
votes
0answers
29 views
Discrete logarithm - strange polynomials
If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number ...
3
votes
2answers
109 views
Polynomial $x^3- xy^3$ and the like over finite fields.
Let $f_{a,n}(x_1,x_2)$ be a polynomial in $\mathbb{F}_p[x_1,x_2]$, where $\mathbb{F}_p$ is a finite field or oder $p$ (perhaps, we may first assume that $p$ is prime) depending on $a\in\mathbb{F}_p$ ...
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vote
1answer
54 views
extended linear codes over the field $\mathbb F_q$
Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where
$$
C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } ...
3
votes
2answers
82 views
Why is $X^4 + \overline{2}$ irreducible in $\mathbb{F}_{125}[X]$?
I want to prove that $f = X^4 + \overline{2}$ is irreducible in $\mathbb{F}_{125}[X]$.
I know that $\mathbb{F}_{125}$ is the splitting field of $X^{125} - X$ over $\mathbb{F}_5$, and that this is a ...
3
votes
1answer
76 views
Finding number of solutions to an equation in $\mathbb F_p$
$p=3 \pmod 4$ is a prime. Let $b\in \mathbb F_p^*$.
Show that the equation $v^2=u^4-4b$ has $p-1$ solutions $(u,v)$ with $u, v \in \mathbb F_p$.
If we write the given equation as $v+u^2=x$ and ...
1
vote
2answers
44 views
Convert polynomials and fractions in a finit field?
I am trying to understand how finite field works, and I am stuck on converting high power polynomials into a power of the field, also converting fractions into integers.
$8^{-1}\cdot44$ in $\Bbb ...
0
votes
0answers
55 views
How to list all irreducible polynomials in a field?
I am currently trying to refresh my memory on some basic primary polynomials, apologies if my terminologies aren't correct:
For example, I have a field $\Bbb F_{2^3}$ and generates a list of ...
2
votes
1answer
53 views
Rank of matrices multiplication
Matrices $m_1$ and $m_2$ are over a finite field($GF(2^{8})$ for example).
$m_1$ is a $m\times n$ matrix($n > m$) with $rank(m_1) = m$,
and $m_2$ is a $n\times c$ ($c > n > m$) matrix ...
