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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An elementary question regarding a multiplicative character over finite fields

Reading Chapter 2 of Koblitz's Introduction to Elliptic Curves and Modular Forms, I got stuck on the following question. I would like to proceed my reading, so I would appreciate any hint to this. I ...
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a square in a finite field of odd order

GF(q) is a finite field of order q, where q is odd. Prove that $a\in GF(q), a\neq0$ has a root in $GF(q)$ iff $a^{(q-1)/2}=1$. I tried to prove it this way: Suppose a has a root in ...
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27 views

Absolute value of a cubic Gauss sum (over Field $\mathbb{F}_p$ )

I'm interested in the quantity $|\sum_{x \in \mathbb{F}_p} \omega^{ax^3+bx^2+cx}|$ where $a \in \mathbb{F}_p^*,b,c \in \mathbb{F}_p$ and $\omega$ is a primitive $p$-th root of unity i.e. ...
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22 views

$2^{p}\equiv1$ mod $2p+1$ for certain $p$

Let $p$ be a prime number such that $p\equiv3$ mod $4$ and $2p+1$ is also a prime number. It is well known that $2^{p}\equiv1$ mod $2p+1$, but I haven't been able to prove it.
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Zeros of $317x^{2}-151xy+40y^{2}$ over $\mathbb{F}_{31}$

Let $K:=\mathbb{F}_{31}$ and $f(x,y):=317x^{2}-151xy+40y^{2}$. I have to find out if there exists any point $(a,b)\in K^{2}$ such that $f(a,b)=0$ and $a\neq0$ or $b\neq0$.
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23 views

Sequences length for LFSR in the general case

An LFSR with a reducible polynomial can generate several sequences, depending on the initial value. My goal is to have an algorithm to compute those length without going through the enumeration of all ...
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1answer
181 views

Understanding a proof of Wedderburn's little theorem

I am working on the proof of Wedderburn theorem and I have a problem to understand the part of it. I don't understand why $b_{1}^{-1}a_{1}=\lambda^{i}$ implies the last contradiction ...
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4answers
79 views

multiplication in Galois Fields

I don't know much about Galois fields. My question is the following Assume we are working with GF(8). Let say for example I want to multiply 2 by 4 in GF(8). Then it should be equal to $2*4 \text{ ...
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Euclid and finite fields

In 300 BC or so Euclid pointed out that if $S$ is any finite set of prime numbers then the prime factors of $1+\prod S$ are not in $S$, so that $S$ can always be extended to a larger finite set. Much ...
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28 views

Subtraction in GF(2^8) Giving Incorrect Results

Let me preface this by stating that I'm not normally a math person, but I'm currently dabbling in finite fields to help wrap my head around certain cryptographic topics (specifically those based ...
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34 views

Sequences length for LFSR when polynomial is reducible

An LFSR with polynomial 1+x4+x5 = (1+x+x2)(1+x+x3) can generate several sequences, depending on the initial value. If I did not made any mistake enumerating them, the sequences length are 3, 7 and 21. ...
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14 views

Determine if matrix D belongs to Vect(A,B,C)

So there are 4 matrices, A, B,C,D. They belong to field F5. Determine if D belongs to Vect(A,B,C). I have pretty much done all the calculations its just i fail to conclude/find the right value for the ...
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1answer
36 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
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diagonalization of a matrix over finite fields

I'm having a problem with determine whether a matrix is diagonalizable over $\mathbb F_{2}$, over $\mathbb F_{3}$, etc. for example, for the following matrix: $$ \begin{bmatrix} 1 ...
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24 views

the number of cube roots in a finite field [duplicate]

Let $p$ be a prime number, and let $\mathbb F_p$ be the field with $p$ elements. How should I choose $p$ such that all elements of $\mathbb F_p$ have cube roots in $\mathbb F_p$?
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Find minimal set of cosets $C$, so that each $2$ vectors in $A_n$ are in one coset in $C$

Let $F_2^n$ be the set of all vectors of length $n$ with values of $0$ or $1$ and $A_n$ = $F_2^n \setminus(11\ldots1)$. Set $A_n$ contains all vectors except one with all $1$s. We can consider cosets ...
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Blocking set for cosets of codimension $2$

In this paper following theorem is proved: If $V$ is vector space of dimension $n$ over a finite field $F$ of $q$ elements then any subset of $V$ which meets every hyperplane of $V$ contains at least ...
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Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?

Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$. Is ...
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zeros of a simple polynomial

For $x, y \in GF[2^n]$, consider the two-parameter polynomial $P(x,y) = x \cdot y + f(x) + g(y)$, where $f$ and $g$ are arbitrary polynomials on $GF[2^n]$. Can we say anything about the number of ...
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How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
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1answer
16 views

system of equations with coefficients in finite field

Suppose we have three simultaneously equations with $4$ variables with coefficients in finite fields, i.e. $$\alpha_1A_1 + \beta_1B_1+\gamma_1C_1 + \theta_1D_1=x$$ $$\alpha_2A_1 + ...
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irreducible quadratic trinomial over finite field

In p130 of 《finite field 》 by Lidl et al. : For a trinomial x^2 +x +a over a finite field F_q of odd characteristic it is easily seen it is irreducible over F_q if and only if a is not of the form a = ...
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Algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
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1answer
34 views

Which powers of a primitive element of a finite field yield a generator of a finite field extension?

Let $F_{q^m}$ denote the finite field with $q^m$ elements. Let $\gamma$ be a primitive element of $F_{q^m}$. What are the powers $i$ such that $F_q(\gamma^i)=F_{q^m}$? Note that the following ...
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35 views

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field.

If $k>0$ is a positive integer and $p$ is any prime, when is $Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in Z_p\}$ a field. Find necessary and sufficient condition Attempt: Since, we know that a finite ...
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384 views

Let F be a field of order 32. Show that the only subfields of F are F itself and {0,1}.

$F$ is a field of order $32$. $F$ and {$0,1$} are trivial subfields of $F$. But how can we show that these are the only subfields of $F$? Can someone give me a direction to this question?
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If $X^{p^d}\equiv X\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$

Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
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Structure of $G$-invariant bilinear forms over (finite?) fields

I have a question about the structure of $G$-invariant bilinear forms. Let $G$ be an arbitrary finite group and $\mathbb{F}_q$ a finite field such that $2|G|$ is not divisible by the characteristic of ...
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What is the algebraic normal form of $F(x,y,z)= Trace (\alpha x^{24}) + x^{312} + yz$?

Let $w$ be a primitive element of $\mathbb F_{5^4}$. Let $\alpha=w^{13}$. Define, $F:\mathbb F_{5^4}\times \mathbb F_{5}\times \mathbb F_{5} \Rightarrow \mathbb F_{5} $ as, $$F(x,y,z)= Tr (\alpha ...
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1answer
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When is $c\alpha$ primitive, for nonzero $c\in GF(q)$ and $\alpha$ primitive in $GF(q^m)$?

Let $q$ prime power and $\alpha \in GF(q^m)$ be primitive element. When is $c\alpha$ still primitive in $GF(q^m)$? EDIT: More generally, if $\alpha \in GF(q^m)$ is any non-zero element, I'm also ...
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Dimension reduction over finite fields

Let $\mathbf b_1, \cdots, \mathbf b_n$ be a basis for $\mathbb F^n_q$ (where $\mathbb F_q$ is a finite field of size $q$). Assume $c \in \mathbb F_q$ is a uniformly randomly chosen number. For a given ...
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Subspace-evasive set performance in the random case

A subspace evasive set is defined as a large subset of a vector space which has small intersection with any $k$ dimensional affine space. That is, it "evades" all affine subspaces of small enough ...
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Operations in finite field $F_p$

If I have a finite field $F_p$, where $p$ is prime how can I define operations like $+, -, \times, / $? Can I just make: $$add: (a + b) \mod p$$ $$sub: (a-b) \mod p$$ $$mul: (a\times b) \mod p$$ ...
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1answer
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How are the fields $\mathbb{F}_k$(where $k$ is an integer) be generated?

What are elements like in the fields $\mathbb{F}_k$? Does $\mathbb{F}_k$ contain only $k$ elements? When $k$ is a composite integer, what will be different from that $k$ is a prime? Please help me.
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Group does not contain any elements of order $p^2$?

I am trying to show that the group of $3 \times 3 $ upper triangular matrices over the field $ \mathbb{F}_p $ with diagonal entries 1 does not contain any elements of order $p^2$ when $ p \geq 3$. ...
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43 views

Bounds for the norm of certain additive character sums

Let $q$ be a power of a prime, let $\alpha$ be a primitive element in the finite field with $q$ elements and let $\chi$ be the canonical additive character of the field. Recently I became interested ...
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Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
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Isomorphism of Groups (C.S.I.R)

Suppose $(F , +, .)$ is the finite Field with 9 elements. Let $G = (F , +)$ and H = (F \ {0}, .) denotes the underlying additive and multiplicative groups respectively, Then $ G \cong \mathbb Z_3 ...
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Approximating polynomials over finite fields

Consider a binary finite field $F = GF[2^{n}]$ with addition and multiplication denoted by $\oplus$ and $*$, respectively. Let me represent the elements of $F$ by $n$-bit strings, which means that ...
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Show two multiplication tables of GF(8) are isomorphic [duplicate]

How to show the two tables above are isomorphic? I try to map one element to another element in another table, but I fail to do so as I found that one element from the table on the left is mapped to ...
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115 views

Polynomial multiplication modulo polynomial

Suppose we are working on finite field $F_{16}$ and have pritimive polynomial $z^4+z+1$. I stuck at how to compute polynomial modulo. For example, we have $z^5+z+1$ mod $z^4+z+1$. I use the usual ...
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1answer
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What is the number of distinct subgroups of the automorphism group of $\mathbf{F}_{3^{100}}$?

Let $G$ denote the group of all the automorphisms of the field $\mathbf{F}_{3^{100}}$ that consists of $3^{100}$ elements. What is the number of distinct subgroups of $G$?
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Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
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Multiplicative group of a finite field

Field $\mathbb{F}$ is finite if and only if its multiplicative group $\mathbb{F}^{\times}$ is finitely generated. The "$\Rightarrow$" implication is obvious, but how to prove the otherwise?
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1answer
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Can the set L, of all even polynomials be a subspace of F[X]?

So I have the question Let F be a field and let L be the set of all polynomials f(x) element of F[X] satisfying the condition that deg(f) is even. Is L a subspace of F[X]? I would say that L is not a ...
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Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? [duplicate]

Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? (It's well-known that if $F$ is a finite field, $F^*$ is a cyclic group). Thank you in advanced.
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3answers
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Why is the multiplicative group of a finite field cyclic? [duplicate]

Why is the multiplicative group $(K\smallsetminus\{0\},\cdot)$ of a finite field $(K,+,\cdot)$ always cyclic?
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1answer
34 views

Existence of “simple” irreducible polynomial of degree 12 in a finite field

Assume that we have a finite field $\mathbb{F}_p$, where $p$ is prime, $p \equiv 1\ (\textrm{mod}\ 4)$ and $p \equiv 1\ (\textrm{mod}\ 3)$. I was looking for irreducible polynomial in a form $X^{12} + ...
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38 views

Show that the following Statements

Consider $F = F_3[x]/(x^3 + 2x -1)$, where $F_3$ is the field with 3 elelments. Then $F$ is either normal extension or inseperable extension. Automorphism gruop of $F$ is cyclic. I know that $x^3 ...
3
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1answer
46 views

Elements of subfield $F_{8}$ of $F_{2}[x]/(x^{6}+x+1)$

I need to find the elements of the subfield $F_{8}$ of $F_{2}[x]/(x^{6}+x+1)$ in their standard representation. I know that $F_{2}[x]/(x^{6}+x+1)$ represents the residu classes of polynomials modulo ...