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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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 ...
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1answer
47 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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24 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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89 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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34 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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27 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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34 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
49 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 ...
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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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33 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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61 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: $$\mathbf{Z}[X],\quad \mathbf{Z}[X]/(X^2+1),\quad \mathbf{Z}[X]/(2,X^2+1)\quad \mathbf{Z}[X]/(2,X^2+X+1)$$ ...
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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 = ...
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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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25 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 ...
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1answer
51 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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1answer
40 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 ...
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20 views

Polarization of a quadratic Hermitian form

Suppose $f$ is a conjugate-symmetric sesquilinear form over a vector space $V$ defined over $F=\mathbb{F}_{q^2}$, i.e. $f:V\times V \rightarrow F$ such that $$f(\lambda u + v, w) = \lambda f(u, w) + ...
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32 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 ...
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51 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 $\mathbb{F}_{5^n}$.
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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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71 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 ...
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150 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 ...
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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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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 $m$?
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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 $2^8$ ...
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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 ...
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1answer
24 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 ...
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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
61 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 ...
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1answer
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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 $X$ ...
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1answer
51 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 ...
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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] / ...
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24 views

Pythagoras number of $\mathbb{F}_p$

For a commutative ring $A$ and $a \in A$, define the length of $a$ as $$ l(a) = \inf \lbrace n \in \mathbb{N} \mid \exists a_1, \ldots, a_n \in A : a = \sum_{i=1}^n a_i^2 \rbrace . $$ Let $\Sigma A^2$ ...
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Selecting an element of order $q$ in $(\mathbb Z/p\mathbb Z)^*$?

Suppose $q\mid p-1$ where $p,q$ are two distinct primes. Also let $h\in[1,p-1]$, then compute $g=h^{(p-1)/q}$ mod $p$. If $g\neq1$, then does it mean that $g$ is of order $q$? If yes, then how? ...
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1answer
39 views

How to write -1 as a square in a finite field of characteristic 2

If $\mathbb{F}_{q}$ is a finite field of characteristic $2$, where $q$ is a power of $2$ and $\beta$ is a generator of $\mathbb{F}^{*}$, then I know that $-1$ is a square in $\mathbb{F}$, but how do I ...
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Show that $x^{p^n}-x$ is the product of all monic irreducible polynomials in $\mathbb{Z}/p\mathbb{Z}[x]$ of a degree $d$ dividing $n$. [duplicate]

I know that if $F$ is a field of $p^n$ elements contained in an algebraic closure $\overline{\mathbb{Z}/p\mathbb{Z}}$ of $\mathbb{Z}/\mathbb{Z}p$, then the elements of $F$ are the zeros $x^{p^n}-x$, ...
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Proving the order of a finite field without using the characteristic

I'm trying to prove the order of a particular quotient field of $\mathbb{Z}[i]$ without using the characteristic of the field. My hesitation in using the characteristic comes from the fact that I'm ...
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2answers
46 views

Algorithm to find the irreducible polynomial

What algorithm can be used find an irreducible polynomial of degree $n$ over the field $GF(p)$ for prime $p$. The reason I ask is I want to make a program for finite field arithmetic, but creating a ...
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Finding the multiplicative inverses in fields

Let's say I have the field $F_{11}$. Why does $2$ have the multiplicative inverse $6$? In some of the examples I have, let's say we are looking $F_5$, why are values up to only $2$ considered? So ...
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1answer
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Product of the norms of two vectors w.r.t a symmetric bilinear form

Let $V=V_{n}(q)$ be a $n$ dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $(,)$ be a symmetric bilinear form on $V$. Fix $v\in V$. I would like to show that there exists a ...