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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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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+100

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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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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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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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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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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33 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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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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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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57 views

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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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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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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1answer
17 views

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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38 views

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
55 views

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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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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1answer
31 views

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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70 views

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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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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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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2answers
113 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
70 views

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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1answer
64 views

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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102 views

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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17 views

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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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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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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35 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 ...
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1answer
23 views

Showing $f\in\mathbb{F}_{p^d}[X]:f'=0\Rightarrow\exists g\in\mathbb{F}_{p^d}[X]:f=g^p$

Let $\mathbb{F}_{p^d}$ denote the final field with $p^d$ elements and $\mathbb{F}_{p^d}[X]$ denote the polynomial ring in $X$ over $\mathbb{F}_{p^d}$. How can we show ...
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Elements with order 3 in group $F_{16}/\{0\}$

If you have the finite field $GF(16)$ and you define the group $GF(16)/\{0\},*$ this group is cyclic. I need to determine how many elements in this group have order 3. Of course you could just try out ...
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45 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 ...
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122 views

An example concerning some fields

I was trying to understand the following example This is also why I've made some questions today on locally finite fields. I've almost understand everything (thanks also to some of you) but I ...
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33 views

Projective special linear group

What is it the minimum number of generators for $PSL(2,\, \mathbb{F}_q)$? Is it known? Is there some references I could see?
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56 views

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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38 views

Minimal polynomial reducible modulo every prime $p$

Suppose $K = \mathbb{Q}(\alpha)$ with $\alpha = a + b\sqrt{D_1}+c\sqrt{D_2}+d\sqrt{D_1D_2}$ with $D_1,D_2 \in \mathbb{Z}$. Prove that the minimal polynomial $m_\alpha(x)$ for $\alpha$ over ...
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fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
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If p is a prime positive integer, find all subfields of GF(p)

If p is a prime positive integer, find all subfields of GF(p) This question just seems too vague.
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If $F$ is a field, show the following function is a permutation

Let $F$ be a field. Show that the function $a\rightarrow a^{-1}$ is a permutation of $F\{0_F\}$ So I know that if it is indeed a permutation, then it is one-to-one and onto. Also, For every $a$,$b$ ...
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33 views

Does the element exist in the Galois Field?

Let p be a prime positive integer and let a be an element of GF(p). Does there necessarily exist an element b of GF(p) satisfying b^2=a? So, taking a element of GF(p), can we find a b element of ...
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Some properties of elements of the finite field GF(q)

Consider the finite field $GF(q)$ and an element $w$ of order $n$ which is an $n$th root of unity in this field. Could someone give me explanation or reference to the following questions regarding ...
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Let $F$ be a field of 8 elements and $A$= {$x\in F$| $x^7$=1 and $x^k$$\neq$1 for all natural number k<1}. Then the number of elements in A is

Let $F$ be a field of $8$ elements and $A$= {$x\in F$| $x^7$=1 and $x^k$$\neq$1 for all natural number $k<1$}. Then the number of elements in $A$ is: a) 1 b) 2 c) 3 d) 6 Please give me some ...
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43 views

The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3 $$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
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Why is the algebraic closure of a finite field countable?

An algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. But why is it a countable set?
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About a field of order $2^{n}$ with $n$ an odd integer and an additional property

I'm new in the world of fields (so I don't have any strong theorem at my disposal) and I've got stuck in this problem: Given a field of order $2^{n}$ with $n$ an odd integer and $a,b$ elements ...
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49 views

Cyclic subgroups of $GL_2(F_q)$

Let $F$ be a finite field with $q = p^f$ elements for $p$ a prime. I know that $G = GL_2(F_q)$ contains a cyclic group of order $q-1$. It is the set of matrices of the form $\begin{pmatrix} x & ...
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1answer
39 views

$x=y^p-y$, $y \in \mathbb{F}_{p^2}$. Prove: $x^2 \in \mathbb{F}_p$

Im practicing for an exam, can anyone help me with this? Let $p$ be prime and let $x=y^p-y$, for $y \in \mathbb{F}_{p^2}$. Prove: $x^2 \in \mathbb{F}_p$
2
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1answer
67 views

Primitive polynomials in LFSRs

I need help proving the following theorem. I found it many books but on every single one it says that they omit the proof because it is in every good textbook. THM Let $c(x)$ be a connection ...
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1answer
50 views

Additive character sum of primitive elements over finite fields

Let $\mathbb{F}_q$ be a finite field with $q$ elements and multiplicative generator $\alpha$, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $N$ be a divisor of $q-1$. ...