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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Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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160 views

Why must a field with a cyclic group of units be finite?

Let $F$ be a field and $F^* \subseteq F$ be its group of units. If $F^*$ is cyclic show that $F$ is finite. I'm a bit stuck. I know that I can represent $F^* = \langle u \rangle$ for some $u \in F^*$ ...
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40 views

Addition in $\operatorname{GF}(2^4)$

How can I compute $A(x)+B(x) \mod P(x)$ in $\operatorname{GF}(2^4)$ using the irreducible polynomial $P(x)=x^4+x+1$. What is the influence of the choice of the reduction polynomial on the computation? ...
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43 views

Uniqueness of projective plane of order 5

Is there a slick way to see the uniqueness of projective plane (equivalently, an affine plane) of order $5$?
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52 views

Linear independence of finite field elements and subfields

Let $q$ be a prime power and $n=lm$ an integer with $l,m>1$. We know that the finite field $GF(q^n)$ is a $n$-th dimensional vector space over $GF(q)$, and it is also a $l$-th dimensional vector ...
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47 views

Symbolic computations in finite fields of unspecified order

The general setting is that I want to multiply some matrices (to many to do it by hands) over a finite field. The problem is that these matrices depend on certain parameters taken from the field and ...
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61 views

Extension field of F2 , expressing roots and primitive elements in that field

Let $\Phi$ be an extension field of $\Bbb{F}_2$ of extension degree s >1. Let $a(x)$ be a non-zero polynomial with the coefficients in $\Bbb{F}_2$. (a) Show that if $\beta$ is a root of the ...
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How to express each element in a field F as a power of a primitive element? [closed]

I have a field F(2^4) and it is represented as a residue ring of the polynomials over F2 modulo the polynomial β4+β3+β2+β+1. I want to express each element in this field as a power of a primitive ...
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1answer
36 views

Is this function part of the Galois group?

Let $\Bbb K$ be a Galois extension of $\Bbb F_p$, where $p$ is prime. Let $\Phi$ be a function on $\Bbb K$. If $\Phi$ is an automorphism of $\Bbb F_p$ that permutes the roots of the minimal ...
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81 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
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53 views

Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
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25 views

multiplication in finite fields irreducible polynomial

I just started doing some reading about multiplication in finite fields and i keep stumbling over one point: in the field G(2^8) how does x^8 + x^4 + x^3 + x + 1 = 0 imply that x^8 = x^4 + x^3 + x + ...
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1answer
34 views

Finite field and Automorphism

Problem 1. Let S be a finite field of characteristics 2 and the map be define as $\eta$: S$\longrightarrow$S x$\longmapsto$x$^p$ Show that $\eta$ is automorphism, i.e., S is isomorphism ...
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53 views

Finite fields formed from irreducible polynomials

Let's say I have an irreducible polynomial $f(x) \in\mathbb Z/(2)[x]$ with a degree n that is at least 2 or greater. How would I go about proving that $\mathbb Z/(2)[x]/(f(x))$ is a finite field of ...
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$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if ...
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Simpler method to show $k(x)$ is NOT a primitive irreducible polynomial?

Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive. Ok, I showed that $k(x)$ is irreducible since it it a quadratic and ...
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93 views

The group structure of elliptic curve over $\mathbb F_p$

I want to find the group of the elliptic curve $y^2=x^3-x$ over $\mathbb F_p$ for all primes $p \le 29$. But I know only 1 fact about the structure of this group: $E(\mathbb F_p)=\mathbb Z/m \mathbb Z ...
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Order Of The Intersection of Two Subfields.

Last question haha, Let $E$ and $F$ be subfields of $GF(p^n)$. If $|E| = p^r$ and $|F| = p^s$, what is the order of $E\cap F$? I read a corollary that "A finite field of order $p^n$ contains a ...
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39 views

Showing an element in a Finite Field can be written as a power.

I had a question that I'm stuck with: Show that every element in $GF(p^n)$ can be written in the form of $a^p$ for some unique $a\in GF(p^n)$. So this field is the splitting field for the polynomial ...
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99 views

Multiplicative subgroup of a finite field with prescribed trace.

Any suggestions/methods/estimates for the following problem would be very appreciated. $l,p$ are primes with $p \equiv 1 \!\! \pmod l$. $r$ is a positive integer with $r \equiv 1 \!\! \pmod p$ and $q ...
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1answer
34 views

Generators of $\mathbb{F}_9/\mathbb{F}_3$ that do not generate $\mathbb{F}_9^{\times}$

Find a generator of the extension $\mathbb{F}_9/\mathbb{F}_3$ that does not generate the multiplicative group $\mathbb{F}_9^{\times}$. how many such elements exist? what are their minimal ...
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55 views

Construction of a polynomial

Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variable prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, monic, ...
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Number of monic polynomials = $q^n$?

In the situation $q=p^k$ with $p$ prime and $k \in \mathbb{N}$ I have the following question: Why is the number of monic polynomials of degree $n$ in $\mathbb{F}_q[X]$ $$q^n \ ?$$
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find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
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153 views

Calculating a strange algebraic limes

I have a problem with calculating a strange limes: Let $ m \in \mathbb{N}$ be fixed and $q=p^n$ (a variabel prime power) for $n \in \mathbb{N}$ and $p$ prime. We define $$c_m=|\left\lbrace f \in ...
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1answer
131 views

Trace of Frobenius of elliptic curve is integer

I recently started to read the book "Arithmetic of Elliptic Curves" by Silverman. And I can't solve an exercise 5.10. Let $E/\mathbb F_q$ be an elliptic curve and $\phi$ is Frobenius endomorphism, ...
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49 views

Multiplicative order in field extension

Let $F/K$ be some field extension (both are finite fields) and $u$ be some element in $F$. I want to know if $u^{|K|} = u$ implies $u \in K$. And why?
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24 views

Finite Field Homomorphism

Let's say that I have a finite field K with characteristic 2. I define @ as a map where @ : K -> K, and x -> $x^2$. First of all, what are some examples of fields like K? I initially thought it ...
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104 views

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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41 views

set of integer modulo m for composite values

Is there a Set of integer modulo m, Zm, where m is composite and the set is a field, (all of its elements having a multiplicative inverse) ? I have heard that the set of integer modulo 4 cannot be a ...
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84 views

Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form ...
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Simplifying presentation of elements of finite field

Let me describe my question through an example. Finite field of order (for example) 8 can be constructed as $\mathbb{F}_8 = \mathbb{F}_2[t]/(t^3 + t + 1)$. So one of a natural presentation of the ...
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23 views

Decision Diffie Hellman in finite fields

Is there an efficient mathematical algorithm for Decision Diffie-Hellman problem in a finite field $F_q$? I have found a detailed analysis of many more involved or specific cases but nothing on the ...
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When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
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87 views

Finite field with algebraic element over $\mathbb{Z}_p$

Let $F$ be a finite field of characteristic $p$. Show that every element of $F$ is algebraic over $\mathbb{Z}_p$. Since char$(F)$ = $p$, $\forall a \in F $ not zero, $ap$ = 0. We also have the same ...
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Understanding Intel's white paper algorithm for multiplication in $\text{GF}(2^n)$?

I'm reading this Intel white paper on carry-less multiplication. For now, suppose I want to do multiplication in $\text{GF}(2^4)$. We are using the "usual" bitstring representation of polynomials ...
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Remainder and quotient of polynomial division in $\mathbb Z_7[x]$

Find the quotient and remainder of: $$ \frac {x^4 + [3]x^3 + [5]x} {x^2 + [2]} ; \mathbb Z_7[x]$$ Would the answer be the same if it wasn't in $\mathbb Z_7[x]$. Namely: Quotient: $x^2 +[3]x - ...
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Group-Isomorphism problem

I want to find an group-isomorphism $$ \psi : (\mathbb{Z}/8\mathbb{Z},+) \longrightarrow \mathbb{F}_9^\times $$ which should be used to multiply elements in $\mathbb{F}_9$ or to find the inverse ...
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Linear Independence for different fields

I have a statement for a space over $R^n$: {x, y, z} is linearly ind. $\implies$ {x + y, x + z, y + z} is linearly independent Quick proof: a(x+y) + b(x+z) + c(y+z) = 0 $\implies$ (a+b)x + ...
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The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
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Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
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A finite sum of $1$ equals $0$ in a field with finitely many elements.

I need to prove that for field $\mathbb{F}$ with finitely many elements $\exists n \in \mathbb{N}$, $\underbrace{1+1+1+...+1}_{n \text{ times }} = 0$. I can see why this is true, since there is a ...
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1answer
63 views

The degree of an irreducible polynomial divides an integer n.

Let $F$ be a finite field such that $|F|=q$, and $f(x)\in{F[x]}$ is irreducible with $deg(f)=d$, $g(x)=x^{{q}^{n}}-x$. Then prove that $f$ divides $g$ if and only if $d$ divides $n$.
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The Frobenius Automorphism

If a finite field $F$ has characteristic $p$, prove that every element $a\in F$ can be expressed as $a=b^{p}$ for some $b\in F$? Hint: Frobenius Automorphism. Isn't the Frob Automorphism about $a$? ...
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Find irreducible polynomials in $\mathbb{Z}_{3} [x]$ having as roots $a + 1$ and $a - 1$

$x^{2} + 1$ is irreducible in $\mathbb{Z}_{3} [x]$, and so K = $\mathbb{Z}_{3} [x] / \langle x^{2}+ 1\rangle$ is a field with 9 elements. Let $a$ in K be a root of $f(x)$. Find irreducible polynomials ...
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Finding a gcd in the form of a monic polynomial

The question is to find the greatest common divisor (in the form of monic polynomial) in $\Bbb F_5[X]$ of $f=x^2-x+4$ and $g=x^3+2x^2+3x+2$ I used the Euclidian Algorithm for polynomials and found ...
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permutation polynomial

If we have GF(4) as an extension field, we can define a permutation polynomial of GF(4) like L(x), a linearized polynomial, of the followinf form: L(x)= \sum_{s=0}^{\r-1} a_s x^(q^r)e Is it possible ...
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47 views

basis for finite fields as a vector space

If we consider GF(4) as a vector space over GF(2), the basis of GF(4) includes two elements 1 and a. Due to this fact that for an arbitrary vector space we can find several basis, what are other bases ...
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78 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
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1answer
52 views

Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...