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 $\operatorname{GF}(q)$.

learn more… | top users | synonyms (1)

3
votes
0answers
39 views

a symmetric matrix over GF(2)

let $A$ be a symmetric $n$ by $n$ matrix over $\mathbb{GF}(2)$. Using elementary linear algebra, it is quite easy to show that diag $A$ is in the range of $A$, where diag $A = [a_{11},a_{22}, \dots, ...
0
votes
2answers
46 views

Polynomials over a field having equal polynomial functions

Let $p(x),q(x)$ be two polynomials over a field $F$ such that $p(a)=q(a)$ for all $a\in F$. Can we say that always $p(x)=q(x)$? If $F=\mathbb Z_5$, then it is possible to find examples of $p(x)\neq ...
0
votes
0answers
23 views

Want to know the best books on Matrix algebra dealing with the special case of over GF(2).

Want to know the best books on Matrix algebra dealing over GF(2). Want to know the special properties of matrices over GF(2). For example, in GF(2), non-zero vectors can have a dotproduct with ...
2
votes
2answers
34 views

Proving that $x^4+x^3+1$ has no solution over $\mathrm{GF}(2^e)$, $e$ odd.

I'm trying to find out whether $x^4+x^3+1$ has a solution over $\mathrm{GF}(2^e)$, $e$ odd. Some quick calculations for small values of $e$ indicate that it does not, but my background is not in ...
1
vote
0answers
63 views

Plotting Elliptic Curve over Finite Field in Maple

Not sure if I might be in the wrong section, but I am looking for guidance on how to plot an elliptic curve over a finite field in Maple. I have tried looking it up but only getting good results for ...
2
votes
1answer
40 views

Easy way to find the order of elements in a finite field

I am trying to work out the multiplicative order of each non zero element in $F_7$. Lets say I am looking at the number $3$. I know its order is $6$. Instead of having to work out the powers of ...
2
votes
1answer
33 views

Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
2
votes
2answers
47 views

Show irreducible polynomial by factorization in $\mathbb{F}_2[X]$ and $\mathbb{F}_3[X]$

I found a problem, I don't really know how to solve, although it should be something very easy, since it is stuff of Algebra I. Let $f= 29X^5−13X^4−44X^3+ 18X^2+ 35X+ 10\in\mathbb{Z}[X]$. 1) ...
1
vote
1answer
140 views

Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$

Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$. Work so far: $$x^8 - x = x(x^7 - 1) = x(x - 1)(x^6+x^5+x^4+x^3+x^2+x+1)$$ From here, I think the Zeros of an Irreducible over a ...
1
vote
2answers
34 views

Discrete math: find prime number which solves the following conditions

Find prime number $p$ and polynomial $f$, so that the ring $F_p[x]/(f)$: 1) contains non-zero element $w$: $w^n=0$ for some $n$ 2) doesn't satisfy 1), but is a field 3) is a field which contains $8$...
0
votes
1answer
54 views

How to find the elements of a finite field?

OK, I am asked to find the elements for the addition and multiplication tables for the finite fields with eight elements. I know I've asked about this question previously, and I've almost gotten a ...
0
votes
0answers
35 views

Cyclic code and its generating polynomial

Can someone give me an example of a $[3,2]$ linear cyclic code and its generating polynomial? Does this mean it is a binary cyclic code with length 3? Also how do I find all binary/ternary cyclic ...
1
vote
0answers
21 views

Kloosterman Sums and K3 Surfaces

In R. Livne's Motivic Orthogonal Two-Dimensional Representations of $\mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, he provides a technique for computing the $\zeta$-function of the $K3$ surface $$X=\{...
1
vote
1answer
48 views

Proving the Existence of Finite Fields By Counting the Number of Irreducibles

I want to prove that given a prime $p$ and a positive integer $n$, there is a finite field of order $p^n$. I want to do it by showing that there is an irreducible polynomial of degree $n$ over $\...
0
votes
0answers
27 views

Question about subfield subcode of a cyclic code

If $C$ is a cyclic code over $\mathbb{F}_{q^n}$, then $C|_{\mathbb{F}_q}$ is a cyclic code over $\mathbb{F}_q$. I know this holds for a linear code, what about a cyclic code? Thanks in advance
1
vote
0answers
28 views

What are the cardinalities of the similarity classes of matrices over a finite field?

Similarity of matrices forms an equivalence relation. The set of linear maps between two finite dimensional vector spaces over a finite field is finite. Is there a combinatorial formula for the ...
4
votes
2answers
100 views

Can every element of a finite field be written as a sum of two non-squares?

We know that any element of a finite field $\mathbb{F_{q}}$ ($q$ odd prime power) can be written as a sum of two squares - is the same true for non-squares? Can any element of a (sufficiently large) ...
-3
votes
1answer
36 views

Finding the multiplicative order of elements in $F_7$ [closed]

In $F_7$, why is $6$ the multiplicative order of $3$ ?
1
vote
1answer
29 views

Weil bound for character sums

I know that the well-known Weil bound for character sums is given by $$\left| \sum_{x \in \mathbb{F}_{q}} \chi(f(x)) \right| \leq (d-1)\sqrt{q}$$ where $\mathbb{F}_q$ is a finite field of size $q$, $\...
0
votes
1answer
64 views

Euclidian algorithm on polynomials in Galois field

I feel so defeated. I need to apply the Euclidian algorithm on two polynomials in GF(16). I already have the answers, I just have no idea on how to divide polynomials with coefficients in the finite ...
0
votes
1answer
42 views

How to find multiplicative orders of all elements in $Z_{p^{2}}$

I am working on some finite fields over $Z_{p^{2}}$ and I want to compute multiplicative orders of all elements in this field. Off the top of my head, I'd say I should multiply each $p^{2}-1$ element ...
-1
votes
2answers
51 views

If K is a finite field, proof that $Gl_n(K)$ is not commutative [closed]

The following property was stated during a lecture in Algebra: If K is a finite field and $n \ge 2$ then $Gl_n(K)$ is a non-abelian finite group. I know how to proof that $Gl_n(K)$ is finite but,...
2
votes
1answer
33 views

Set of vectors on $\mathbb{F}_2^n$

Let $v_1, \dotsc, v_N$ be a set of vectors of $\mathbb{F}_2^n$ which has the following property: for any choice of $1 \leq j_1 < \dotsb < j_n \leq N$, the vectors $v_{j_1}, \dotsc, v_{j_n}$ are ...
0
votes
0answers
23 views

How to find all the subfields of $\mathbb{F}_{p^r}$ [duplicate]

How does one find all the subfields of the finite field $$\mathbb{F}_{p^r}$$ where $p$ is a prime?
1
vote
2answers
38 views

significance of no multiple roots $x^q-x$

From Wikipedia on Finite Fields: "The polynomial $X^q-X$ factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polynomials of degree one ...
5
votes
2answers
64 views

Is $\mathbf{Z}[X]/(2,X^2+1)$ a field/PID?

I've been asked to determined whether the following are fields, PIDs, UFDs, integral domains: $$\mathbf{Z}[X],\quad \mathbf{Z}[X]/(X^2+1),\quad \mathbf{Z}[X]/(2,X^2+1)\quad \mathbf{Z}[X]/(2,X^2+X+1)$$ ...
0
votes
1answer
30 views

Galois: is $f(x)$ irreducible in $\mathbb{F}_5$

$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\F}{\mathbb{F}}$I have a function $f(x) = 2(x^2-x)+1$ and my question is, if this is irreducible in $\F_5$. I now that $\F_5$ comes from $5=q=p^n$, $p = \Z/...
0
votes
2answers
49 views

Proving $\mathbb{F}_p[x]/\langle f(x)\rangle$ with $f(x)$ irreducible is a finite field.

Our text makes a statement that by using the Euclidean Algorithm, it can be shown that $\mathbb{F}_p[x]/\langle f(x)\rangle$, with $f(x)$ being an irreducible polynomial of degree $m$, is a finite ...
2
votes
3answers
32 views

Showing that in $\mathbb{F}_q$, $q$ is odd, that $x^2=1$ has two solutions.

So I am having a bit of an issue with two parts of a question: Let $\gamma$ be a primitive element of $\mathbb{F}_q$, where $q$ is odd. Show that the only solutions to $x^2=1$ are $1$ and $-1$, and ...
0
votes
0answers
29 views

maximum number of homogeneous linearly independent equation over $\mathbb{F}_{2^m}$

In Field $\mathbb{F}_{256}$ we are given homogeneous equations in 255 variables from the field. It is said that maximum number of linearly independent such equations we can get are 247. Why ? To be ...
0
votes
1answer
61 views

Question about finding smallest field given a primitive root of unity

I am having a bit of an issue with an example I was looking at. The question states: "What is the smallest field of characteristic 2 that contains a primitive 11th root of unity?". I am not familiar ...
1
vote
1answer
47 views

How to write a given element of the orthogonal group as a product of reflections

Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given ...
0
votes
0answers
34 views

How do I find the multiplicative inverse of a finite field polynomial using Euclidean Algorithm?

How would I use the Extended Euclidean Algorithm to find a multiplicative inverse of a polynomial in a Galois/Finite field...say, for example, $GF(2^3)$ ? Is it the same process as how it's ...
-1
votes
1answer
52 views

Find $n$ such that $f$ is reducible or give a proof. [duplicate]

Let $f = X^4 + 2$ and let $n$ be odd. I am looking for a proof or a counter example to show that $f$ is irreducible or reducible over $\mathbb{F}_{5^n}$.
0
votes
0answers
25 views

Analogue of orbit counting formula in $F[x]$

We know that if $G$ is a finite group acting on a finite set $\Omega$, then $$\sum_{g\in G}\chi(g)\equiv 0\pmod{|G|}$$ where $\chi$ is the permutation character. I wonder if is there an analogue of ...
1
vote
1answer
26 views

Adjoining a root of $p$ to $F_p$ is etale?!

I'm confused about etale extensions of $F_p$. We know the etale extensions of a field are the products of separable finite field extensions. But if you take $F_p$ and adjoin a root of p this is ...
1
vote
2answers
45 views

Show that $\mathbb{F}_{p^2}$ has an 8th root of unity

Let $p$ be an odd prime number. I want to show that $\mathbb{F}_{p^2}$ has a primitive 8th root of unity $\zeta$. I know that $\zeta^8 = 1$. So my idea is to define $f = X^8 - 1$ such that $\zeta$ ...
1
vote
0answers
73 views

Pictures of curves over finite fields with many points

At the manypoint page for $2^3$, genus=3, there is the note: "In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus-3 curve with 24 points over $F_8$: $(x + y +...
3
votes
0answers
152 views

Irreducibility of a polynomial modulo infinitely many primes

Suppose $f(x) \in \mathbb{Z}[x]$ is an irreducible monic polynomial over $\mathbb{Q}(\alpha)$ of degree $n$, where $\alpha$ is a root of a monic polynomial $g(x) \in \mathbb{Z}[x]$. Assume that the ...
3
votes
0answers
42 views

Number of square-free polynomials over a finite field - a combinatorial interpretation?

One can show using zeta functions that the number of (monic)square-free polynomials of degree $n$ over a finite field $\Bbb F_q$ is $q^n-q^{n-1}$. For instance, this is done here. The answer is ...
2
votes
1answer
54 views

When is 2 a quadratic residue in a finite field?

If $F$ is a finite field of order $q$, where $q$ is an odd prime power, then when is $2$ a quadratic residue in $F$? I know the result for when $q$ is prime. I also know a theorem which says that: ...
0
votes
2answers
48 views

The number of primitive polynomials of degree $m$ over a finite field $GF(p^m)$ [closed]

Why is it that over $GF(p^m)$ there are exactly $\phi(p^m − 1)/m$ primitive polynomials of degree $m$?
0
votes
0answers
21 views

Determine how many poly of certain degree are irreducible. [duplicate]

I am trying to calculate how many polynomial of degree 8 in $\mathbb{Z_2}[x]$ are irreducible. A polynomial is irreducible if it can be factored into two polynomials of lower degree. There is $2^8$ ...
0
votes
1answer
33 views

Deriving field 'tables'

Reviewing for a term test and just confused about how these tables were derived. While I'm pretty sure I know what a field is (a set that satisfies those specific field axioms), that table for ...
2
votes
1answer
25 views

Finite Fields — the image of certain subfields under a given norm map

Let $p$ be a prime. Let $n$ and $k$ be positive integers such that $k$ divides $pn$ but not $n$ (that is, $k$ is a divisor of $pn$ having $p$-adic order that is one greater than the $p$-adic order of $...
1
vote
2answers
52 views

Finding isomorphisms between finite fields.

I'm having trouble understanding how to find isomorphisms between finite fields. In my lecture notes it uses the following theorem: A function $f$ is an isomorphism from $GF(z^n)$ represented ...
2
votes
1answer
67 views

Does any polynomial with integer coefficients split over some prime field?

Assume that $Q(x)$ is a polynomial with integer coefficients. Is there a prime number p such that the equation $Q(x)=0$ has all its root in the finite field $\mathbb{Z}/p\mathbb{Z}$? I asked ...
1
vote
1answer
41 views

Does two's complement arithmetic produce a field isomorphic to $GF(2^{n}$)?

From what I understand, we have these two isomorphisms: $(TC, +)$ is isomorphic to the cyclic group $\mathbb{Z}/2^n\mathbb{Z}$. $(TC, *)$ is isomorphic to the multiplicative group of polynomials. ...
6
votes
4answers
143 views

Prove that there is no element of order $8$ in $SL(2,3)$

Let $SL(2,3)=SL(2,\mathbb{F}_3)$. Prove that there is no element of order $8$ in $SL(2,3)$. My attempt: Let $A$ be a matrix in $SL(2,3)$. Then $A=U X U^{-1} $ for some invertible $U$ where $X$ is ...
1
vote
1answer
54 views

Constructing $F_4$ [duplicate]

To construct $F_4$, why do we take $X^2+X+1$? I understand that this polynomial is irreducible $F_2$, what does irreducible mean? And why does $F_2$ come into it? I see $F_4=\{a+b\omega | a,b \in ...