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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Factorization of $x^5-1$ over $F_{11}$ and $F_{19}$

I got the following question. The polynom $x^5-1$ should be factorized over $F_{11}$. I have this as a first solution: $x^5-1=(x-1)(x+1)(x^2+1)(x-1)$ could this be possible? I don't know which ...
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Does this equation in $\mathbb{Z}_p$ always have a solution?

Is there always a solution to the equation in the field $\mathbb{Z}_p$ ($p$ being a prime number) $$ a^2 + b^2 \equiv c \pmod p $$ for a given $c \in \mathbb{Z}_p$? The solution need not be unique, ...
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Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
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Bijections of a finite field that preserve the kernel of the trace

Let $q=p^n$, for some prime $p$ and positive integer $n$. Let $m$ also be a positive integer, and denote $Tr$ to be the trace of $GF(q^m)$ over $GF(q)$. I have the suspicion that all the functions ...
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How to find all possible polynomials over a given finite field?

How would I find the possible polynomials over GF(p)? I'm trying to figure out which polynomials of a specific given finite field have no roots.
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Matrices such that their sum equals their product

Let $N > 1$ and consider square matrices of size $N$. Let $J$ be the matrix full of $1$'s. Suppose you have $n \geq 1$ commuting matrices $A_i$ over some finite field and such that $\sum_i A_i = ...
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Why do the cyclotomic cosets have the same size?

So I have this problem: when $p=2$ and $n=2^m-1$ show that $|C_1|=|C_3|=m$ where $m\geq 3$. $C_s=\{s,ps,p^2s,\dots p^{m_s-1}\}$ is the cyclotomic coset. I know that the size of the coset is $m_s$ ...
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To Factorize $x^{27}-x$ over $\mathbb F_3$.

Problem 7.5 in Chapter 15 of Artin's Algebra asks to factorize $x^{27}-x$ over $\mathbb F_3$. Here is what I have done. $x^{27}-x=x(x^{26}-1)= x(x^{13}-1)(x^{13}+1)$. In am having trouble ...
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Proving that $X^4+1$ is reducible over all finite prime fields

In that article, I prove that the polynomial $X^4+1$ is reducible over all finite prime fields of odd characteristic. The proof is based on the fact that for $p$ odd prime, the multiplicative group ...
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Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?

How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well? I know how to show that it is an integral ...
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Zero as a repeated permental root for a matrix over a finite field

All, Suppose $A \in Mat(n, \mathbb{F}_{q})$ for $q$ prime, $q \geq 5$, and $n \geq 2^{q-2}$ . Let $\pi_A(x)$ be the permanental polynomial for $A$. That is, \begin{equation*} \pi_{A}(x)=per(xI-A). ...
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Counting couples of square-free polynomials over finite fields

I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$: $$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\ y_2^2=h_2(t) ...
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Automorphism group of the general affine group of the affine line over a finite field?

I am wondering what the structure of the automorphism group of the general affine group of the affine line over a finite field looks like. I'll make that a bit more precise: If $k$ is a finite field, ...
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What is known about the numbers $M_p = \left\vert C(\mathbb{F}_p )\right\vert$?

There is a question (2.4.c) marked ** (to denote "extremely difficult/currently open problem") in Silverman and Tate's Rational Points on Elliptic Curves which I found really interesting and wondered ...
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In cryptography, why do we reduce elliptic curves over finite fields?

What's wrong with real numbers? Is the continuous logarithm problem "easy" to solve for elliptic curves? Here's what I believe: elliptic curves over the real numbers have infinitely many points, many ...
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Is $K\mathbb P^n\cong K^n\mathbin{\dot\cup}K\mathbb P^{n-1}$ for all fields $K$?

I know that for $K=\mathbb R$, the statement $\mathbb R\mathbb P^n\cong\mathbb R^n\mathbin{\dot\cup}\mathbb R\mathbb P^{n-1}$ (where $\cong$ denotes set isomorphism) holds. Is the identity $$K\mathbb ...
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In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?

I know that in infinite fields, such as $\mathbb{C}$, the mapping $e^x$ is a homomorphism from the additive group to the multiplicative group, and I was just wondering if in any finite field, there ...
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117 views

Prove that $x^{p^n} -x +1$ is irreducible over $\mathbb{F_p}$ only when n=1 or n=p=2

Question:Prove that $x^{p^n} -x +1$ is irreducible over $\mathbb{F_p}$ only when n=1 or n=p=2 I know it is a duplicate question. However, someone gave some nice hints on this problem and I want to ...
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Mapping the additive group of a finite field of order $2^n$ to its multiplicative group

In a finite field $F$ of order $2^n$, we know that its additive group is isomorphic to $(\mathbb{Z}_2)^n$. We also know that $(\mathbb{Z}_2)^n$ can be thought of as the set of all $n$-digit binary ...
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Binary Goppa Codes: Calculating code characteristics (traces, length L, distance d, k)

I am having (many) troubles with binary Goppa Codes. My question is at the moment: How do I calculate the trace on given points $tr(\alpha^u)$? For example: Given a finite field $GF(2^6)$ and the ...
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Showing that the fixed points of a homomorphism form a finite field.

I have the following question on my problem set. Suppose that $\textsf{k}$ is a field and $\phi:\textsf{k}\to\textsf{k}$ is a homomorphism. Check that $\textsf{k}^\phi=\{x\in\textsf{k}:\phi(x)=x\}$ ...
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Irreducible Polynomials $GF(2^4)$: Why is $x^4 + x^2 + 1$ reducible?

I am currently working with $GF(2)$, in particular with $GF(2^4)$. One task is to find all irreducible polynomials. I have found ways of reducing the list of all candidates drastically. In my current ...
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21 views

Boolean Polynomial Mutliplication modulo an irreducible polynomial

I am currently reading a paper on the Mathematics of RAID 6 by Peter Anvin and cannot get my head around the notation or results used describing multiplication by {02} (hexadecimal). Here is how he ...
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Distribution of Trace values

I try to prove that ${2^{n-1}}$ elements of the field $\mathbf{F}_{2^{n}}$ have a Trace with value 1, while the other ${2^{n-1}}$ elements have a Trace with value 0. I started to show that Trace(1) ...
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50 views

Irreducibility of polynomials over finite field

If I want to show that a certain polynomial is irreducible over a finite field, which methods do I have? In particular how can I show that $X^4-3$ is irreducible over $\mathbb F_5$ The idea which I ...
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resultants over finite fields vs resultants over their closure

I'm having a little trouble grokking the following line of argument. I have two monic, univariate polynomials $p(x)$ and $q(x)$ over $\mathbb{Z}/p\mathbb{Z}$ that share a common root, where $p$ is ...
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Show that if $n$ is not prime then $\mathbb{Z}/n\mathbb{Z}$ is not a field

I read book of Dummit and Foot Abstract algebra. I need some help with the following question. Show that if $n$ is not prime then $\mathbb{Z}/n\mathbb{Z}$ is not a field I know definitions of ...
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determining the multiple solutions for GF(2) discrete logarithms of polynomials with partially known coefficients

I have an LFSR, essentially $x^k \mod p(x)$ for some characteristic polynomial with coefficients in GF(2), as outlined in Clark and Weng's article: it has a period (= order of the associated finite ...
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Generating elements of a Galois Field using an irreducible polynomial

I am practicing some cryptography problems and I am having problems with one involving Galois Fields and irreducible polynomials. Here is the problem: Using the irreducible polynomial $f(x) = x^5 ...
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When does a field of for which a set of matrices is defined on become important?

I have already gone through several linear algebra related courses without any notion of a field and I am having trouble reading a serious linear algebra book that talks about infinite field, so and ...
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Value of a Moebius sum

Let $q$ be a power of a prime, let $n \in \mathbb{N}$. Is the value of the following sum known? $$ \sum_{d \mid q^n-1} \mu\left( \dfrac{q^n-1}{d} \right) q^{ord_d(q)}, $$ where $ord_d(q)$ denotes ...
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What is the general property a field in which addition can be “undone” by addition

For example, in GF(2) we can do the following: p = 110 k = 010 c = p+k = 110+010 = 100 If we want to calculate p or ...
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Construction of a field with $8$ elements.

Could someone tell me if one can build a field with $8$ elements?
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Picking codewords that are close

Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and number of minimum weight codewords $N_d$. How many ways can you select codewords $c_1,\dots,c_T$ (assume $T\ll q^k$) such ...
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When is a generalized Vandermonde matrix over a finite field invertible?

The generalized Vandermonde matrix that I am considering is one where the rows of a matrix correspond to the powers of different elements of the field, but the powers need not be consecutive integers ...
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Zeros of the Ramanujan sum for finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements, let $N$ be a positive divisor of $q-1$, and let $\xi_N$ be an element of $\mathbb{F}_q^*$ of order $N$. One can similarly define the Ramanujan ...
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$F$ a field and $G$ finite subset of $F \setminus \{0\}$ with 1 & satisfying $a, b ∈ G$ then $ab^{−1} ∈ G$. Show that $G$ is cyclic [duplicate]

Let $F$ be a field and let $G$ be a finite subset of $F \setminus \{0\}$ containing $1$ and satisfying the condition that if $a, b ∈ G$ then $ab^{−1} ∈ G$. Show that there exists an element $c ∈ G$ ...
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59 views

Vector spaces over Z/2Z,countably many predicates added,invariant

Suppose we have a class of vector spaces over $Z/2Z$ with predicates $P_n,n\in N$ added for independent subspaces of co-dimension 2. (BTW, independent in which sense,$P_n$ is independent with ...
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Finding a polynom to construct finite field

I'm currently working on the following task: "State a polynom $f(x)\in F_4[x]$ with which you can construct $F_{2^6}$ as $F_4[x]/fF_4[x]$." What I know is, that a polynom $f$ is needed, which must ...
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Upper bounds for the dimension of a binary cyclic code

Let $\mathbb{F}_2 = \{0,1\}$ denote the field with two elements. Consider a binary $N$-tuple $a = a_0 a_1 \ldots a_{N-1}$, of elements $a_i \in \mathbb{F}_2$. The cyclic code $C_a$ corresponding to ...
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Factoring polynomial in prime field

How is a polynomial like $x^5-1$ be factorised in a prime field like $\mathbb{F}_{11}$ for example ? Any advice ? I was successful in trying all members of $\mathbb{F}_{11}$ to find the roots as ...
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Calculate the trace of all elements in $F_8$

I got the following exercise where you have to calc the trace of all elements in ${F_8}$ which is constructed as ${F_2}[x]$/(${x^3+x+1}$)${F_2}[x]$. Up to now I did those steps: 1) Find all elements ...
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Irreducible polynomials of degree $2$ over a finite field

By consider the nonquadratic residues, one has an irreducible polynomial of degree $2$ over ${\Bbb F}_p$ for $p$ being odd: $x^2+r$. Also we know that $x^2+x+1$ is irreducible over ${\Bbb{F}_2}$. How ...
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Factorization of $x^8-x$ over $F_2$ and $F_4$

How can I factorize $x^8-x$ over the fields $F_2$ and $F_4$?
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Proof of lemma about product of irreducible polynomials over finite fields

Reading this Wikipedia page while learning about polynomials over finite fields, I came upon the following lemma: For $i ≥ 1$ the polynomial $x^{q^i}-x \in \mathbf{F}_q[x]$ is the product of all ...
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Bound for degree of polynomial in finite field in order to be solved by common algorithms

For fixing ideas, suppose that we have a field $K$ of $p^n$ elements with (e.g.) $p=7$, $n=55$. Let $f(x)\in K[x]$. What is the maximum degree that $f$ can have in order to be solved by a common and ...
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Proof of Chevalley–Warning theorem

How to prove Chevalley–Warning theorem (http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem) by using Fulton's trace formula (#$|X(\mathbb{F}_p)| \equiv \sum (-1)^i Tr(Frob_p|H^i(X, ...
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The most general splitting of a field extension

Take $L/K$ an extension of the field $K$. I have questions on how we can "split" the extension $L/K$. (1) One knows that $L$ has a transcendence basis $(x_i)_{i\in I}$ over $L$, so that if $E := ...
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64 views

Counting the number of $\mathbb{F}_q$ points on a homogeneous polynomial

This is an area of number theory that I am not too familiar with and I would appreciate any assistance! Let $\mathbb{F}_q$ be a finite field of $q$ elements with characteristic not 2 or 3. I have the ...
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1answer
41 views

Why is $\sqrt{5}$ an element of every field of order $p^{2 e}$?

This was claimed in an answer to another question I asked but it's unclear to me why it's true. I'd also be happy with a reference that explains it. Thanks!