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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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 ...
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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 ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 $8$...
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52 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 ...
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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 ...
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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=\{...
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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 $\...
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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
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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 ...
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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) ...
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36 views

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

In $F_7$, why is $6$ the multiplicative order of $3$ ?
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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 $q$, $\...
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1answer
60 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 ...
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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 ...
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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,...
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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 ...
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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?
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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 ...
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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: $$\mathbf{Z}[X],\quad \mathbf{Z}[X]/(X^2+1),\quad \mathbf{Z}[X]/(2,X^2+1)\quad \mathbf{Z}[X]/(2,X^2+X+1)$$ ...
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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/...
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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 ...
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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 ...
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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 ...
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60 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 ...
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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 ...
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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 ...
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1answer
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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 $\mathbb{F}_{5^n}$.
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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 ...
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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 ...
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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 $\zeta$ ...
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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 +...
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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 ...
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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 ...
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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: ...
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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 $m$?
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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 $2^8$ ...
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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 ...
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1answer
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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 $...
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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 ...
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1answer
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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 ...
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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. ...
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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 ...
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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 ...
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Related Forms for the Riemann Hypothesis over Finite Fields

There are several formulations and consequences of the Riemann Hypothesis for Curves over Finite Fields. I am interested in the logical implications between those, and in elementary (as possible) ...
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Finding the number of irreducible quadratics in $\Bbb Z_p[x]$, where $p$ is a prime [duplicate]

The problem is to find the number of irreducible quadratics in $\Bbb Z_p[x]$, where $p$ is a prime number. To solve this, I wish to find first the number of reducible quadratics of the form $x^2+ax+b$...
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Construct a field of 25 elements.

We need to construct a field of 25 elements. By using a result : For a prime p and a monic irreducible polynomial $p(x)$ in $\Bbb F_p [x]$ of degree $n$ , then the ring $(\Bbb F_p [x] / <p(x)>)$...