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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multiplication in GF(256) (AES algorithm)

I'm trying to understand the AES algorithm in order to implement this (on my own) in Java code. In the algorithm all byte values will be presented as the concatenation of its individual bit values (0 ...
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Solutions of $ 0 = x^2 -ay^2 -1$ in $\mathbb F_q$ where $a$ is not a square.

Assume $F = \mathbb F_q$ where $q = p^r$ for $p$ prime and $r > 0$. I have to count $$ \{(x,y) \in F^2 \mid x^2 -ay^2 -1 = 0\} $$ where $a$ is not a square in $F^*$. The equation is equivalent to ...
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counting symmetric nilpotent matrices

In a recent paper [ Counting symmetric nilpotent matrices , by A. Brouwer], the author states that the number of 3x3 symmetric nilpotent matrices over the field of q elements is given by the ...
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$\mathbb F_9 = \mathbb F_3(i)$ Question about squares

Is it true that $\forall a,b \in \mathbb F_9$: $a \cdot b$ is a square $\iff$ $a \cdot \overline b$ is a square ?
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Factor $x^5-1$ into irreducibles in $\mathbb{F}_p[x]$

I have to factor the polynomial $f(x)=x^5-1$ in $\mathbb{F}_p[x]$, where $p \neq 5$ is a generic prime number. I showhed that, if $5 \mid p-1$, then $f(x)$ splits into linear irreducible. Now I ...
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How to convert a polynomial in $GF(p^n)$ into the form $a(x)^k$, with $a(x)$ a generator polynomial.

I basically have two questions: Given $GF(p^n$) and $g(x)$ an irreducible polynomial: My textbook says that a polynomial is called primitive if $x$ is a generator of the field. The question now is, ...
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Genus over finite fields

Is there a way of computing the genus of a parametrized curve over a finite field? For instance I am interested in the genus of the following space curve in the m-dimensional space over $F_{q^k}$ ...
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Understanding examples of subfield and prime subfield of a finite field

I have already taken a look at this answer. Somehow it did not answer my question. As I can find, in various literatures, A lecture note, Definition 4.1: Let $F$ be a field. A subset $K$ that is ...
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107 views

Finite fields as splitting fields

hey guys so i stumbled upon an example that begins with "Consider GF(25). This can be constructed as the splitting field of $t^2 - 2...$" But the theorem states that it is the splitting field of the ...
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Which accompanying text do you suggest on these topics of finite fields?

Please take a look at pages 80-85 (Section 2.6 Finite fields) of this handbook, Handbook of Applied Cryptography. I am trying to learn the mathematics enumerated in these pages. I do not need the ...
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167 views

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does ...
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Extending and inducing irreducible representations

I sense this may be a simple question, but it is one I haven't been able to find an answer for, possibly due to the use of different terminology. Referring to this question: Faithful irreducible ...
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62 views

What is the smallest dimension possible for a representation of $D_8 \times Q_8$ which is faithful over $F$?

Consider $D_8 \times Q_8$, where $D_8$ is the dihedral group of order 8; $Q_8$ the quaternions. Let $F$ be a field of characteristic not equal to 2. What is the smallest dimension possible for a ...
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Is the splitting field equal to the quotient $k[x]/(f(x))$ for finite fields?

maybe that's an idiot question. Given a finite field $k$ and some irreducible polynomial $f(x) \in k[x]$, then $k_f \cong k[x]/(f(x))$? I know that it's true if $k$ is the prime field and I think that ...
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283 views

Relationship between number of conjugacy classes and number of irreducible representations of a group

For a finite group G the number of irreducible representations over an algebraically closed field F is at most the number of conjugacy classes whose sizes are coprime to the characteristic of F. What ...
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366 views

Degrees of faithful irreducible representations of $\mathbb{Z}_n$ over finite fields

I came across the following theorem while studying representation theory over finite fields. A cyclic group $\mathbb{Z}_n$ has a faithful irreducible representation of degree $d$ over $\mathbb{F}_p$ ...
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187 views

What is the difference between multiplicative group of integers modulo n and a Galois Field

What is the difference between multiplicative group of integers modulo n and a Galois Field? Is $\mathbb{Z}^*_p$ with p prime the same as $GF(p)$? Or is it the same as $\mathbb{Z}/n\mathbb{Z}$? ...
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Primes for which 2 and -2 are residues.

I know that 2 is a residue of primes of the form $8n+1$ and $8n+7$ and so on. I want to find a purely group theoretic or field theoretic proof of these statements. For example, for 8n+1, the ...
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574 views

Prove that the group, under multiplication, of all nonzero elements in a finite field must be a cyclic group.

Prove that the group, under multiplication, of all nonzero elements in a finite field must be a cyclic group. This is what I did, but I'm not sure if it's right: First, we look at the additive ...
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222 views

Isomorphism between two finite fields

We have $k_1:= \mathbb F_7(\alpha)$ and $k_2 := \mathbb F_7(\beta)$ where $\alpha^2 = 3$ and $\beta^2 = -1$ in $\mathbb F_7$. I have to show that these two are isomorphic. Let $\phi:k_1 \rightarrow ...
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Spliting Field over $\mathbb{F}_3$

How to find the splitting field of $f(x)=x^3-x+1$ and $g(x)=x^3-x-1$ over $\mathbb{F}_3$ and how to construct a isomorphism between them?
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finitely generated subfield of algebraic closure of the finite field with $p$ elements

Let $\mathbb{F}^{\operatorname{alg}}_p$ be the algebraic closure of the finite field with $p$ elements. I know that any finitely generated subfield of $\mathbb{F}^{\operatorname{alg}}_p $ is ...
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168 views

Definition of a primitive polynomial

I understand there are already some questions (A, B) on primitive polynomials. But none of these clears my confusion. In page 84 of Handbook of Applied Cryptography, primitive polynomial has been ...
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Construction of the projective plane over $\mathbb{F}_3$

I have a question about constructing projective plane over $\mathbb{F}_3$. We first establish seven equivalence classes $P= \{ [0,0,1], [0,1,0], [1,0,0], [0,1,1], [1,1,0], [1,0,1], [1,1,1] \}$. ...
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Chaos in finite field

Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map $x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $ where $\mathcal{P}$ - ...
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244 views

If $F$ is a finite field of size $q=p^n$ and $b\in K$ is algebraic over $F$ then $b^{q^{m}} = b$ for some $m > 0$

I want to apply a similar type of argument to show that $\alpha^{q} = \alpha$ for all $\alpha\in F$. When we know that the characteristic of $F$ is $p$ and $|F| = q = p^{n}$. But I dont know how to ...
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Basic concepts in finite fields

I need some help with clearing up some some basic concepts in finite fields. I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to ...
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Linear polynomials of finite fields

I have a final tomorrow, and I was looking over some exercises in my textbook. However, I can't seem to work this problem out. Let $F$ be a field of $p^n$ elements and let $\alpha \in F^*$, where ...
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Irreducible polynomials have distinct roots?

I know that irreducible polynomials over fields of zero characteristic have distinct roots in its splitting field. Theorem 7.3 page 27 seems to show that irreducible polynomials over $\Bbb F_p$ have ...
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Irreducible Polynomial in $\mathbb F_{256}$.

Let $\mathbb F_{256}$ be the finite field with $2^8 = 256$ elements. Consider the polynomial over this field $$ x^2 + x + 1. $$ I wanted to know if it is irreducible, so I calculated it for all ...
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Irreducible polynomial roots and representations for Galois field elements in normal basis

I am trying to understand some topics in the literature and came across the following problem. Say I have a field $GF(2^4)$ defined by the irreducible polynomial $r(z) = z^4 + z + 1$, and I want to ...
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Elements of the form $aX^2 + bY^2$ in a finite field.

For cardinality reasons, we know that every element in a finite field $F$ is a sum of two squares. If I fix $a,b\in F$ with $a,b\neq 0$, can every element in $F$ be written in the form $aX^2 + bY^2$ ...
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equality between the index between field with $p^{n}$ elements and $ \mathbb{F}_{p}$ and n?

can someone explain this? $ \left[\mathbb{F}_{p^{n}}:\mathbb{F}_{p}\right]=n $
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Combinatorics in finite vector space

Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$. Let $k$, $a$ and $b$ be non-negative integers. Determine the number of subspaces $K$ of $V$ ...
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Do we really need polynomials (In contrast to polynomial functions)?

In the following I'm going to call a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication) that has the form ...
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Geometric solution to classic committee problem

Most people know the classic committe style problems. I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works. I was hoping someone ...
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Are all of the irreducible representations of (any) symmetric group over $\mathbb{C}$ also irreducible over a finite splitting field.

This is probably an incredibly stupid question, but I'm a novice to representation theory and finite field theory, as I've just been introduced to these concepts, so all I really need is confirmation ...
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63 views

Finding squares in finite fields

I need the definition of finding squares in finite fields and also the number of squares in a finite field. How can we find squares in $\Bbb F_5$ and $\Bbb F_7$? (Here $\Bbb F_5$ and $\Bbb F_7$ ...
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Order of the unit group of a finite field F if for all two subgroups of F one is contained in the other.

Let $F$ be a finite field. Prove that the following are equivalent: i) $A \subset B$ or $B \subset A$ for each two subgroups $A,B$ of $F^*$. ii) $\#F^*$ equals 2, 9, a Fermat-prime or $\#F^* -1$ ...
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Solve equations in a field with characteristic p.

Let $p$ be a prime and $K$ a field with characteristic $p$. How to solve the equation $x^2+2=0$ in the field $K$? Here $x, 2, 0 \in K$. Is it equivalent to solve the equation $x^2+2 = 0 \pmod p$? What ...
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Solve $x^3+x \equiv 1 \pmod p$

Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions. We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$ We can prove that $(1)$ cannot ...
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Seeing that $\Bbb F_2[x]/(x^2+x+1)$ is a field

I have some basic question with polynomials appreciate if someone could explain me this. Following is additional and multiplication tables and it is say that this is a field. Have no idea why say ...
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Full Rank Matrix with a specific construction

Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$ $$Z=\begin{bmatrix} w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
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Is there a pattern of the factorization of a polynomial modulo $p$ as $p$ varies

Take $P\in\mathbb{Z}[X]$ and factorize it modulo $p$, where $p$ is a prime. Modulo different $p$'s the factorization varies. Is there a pattern in this variation? I mean, for example, if $P$ is ...
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Irreducible polynomial over $\mathbb Z/7\mathbb Z$

Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
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Show that for a finite field $F$ and finite group $G$, $F$ is a splitting field for $FG$

Let $F=\mathbb{F}_q$ be the field of $q$ elements and $G$ be a finite group. I'm trying to show that for an irreducible $FG$-module $V$, we have $\mathrm{End}_{FG}(V)=F \cdot 1$, i.e. that $F$ is a ...
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114 views

Finding quadratic residues in a finite field by using a primitive element

Let $1+2x$ be a primitive element of the field $\mathbb F_9$ obtained via the irreducible polynomial $$x^2 + 1$$ over the base field $\mathbb F_3$. i) Make a list of the elements of $\mathbb F_9$ ...
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On any finite field, adding the identity element a finite amount of times will result to the neutral element

Show that if you add the identity element ($1$) a finite amount of times will result to the neutral element ($0$). I started with saying that in every field there's an element $a \in F$ so that $a + ...
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354 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
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Order of matrices in $SL_2({\mathbb{F}_q})$

Could you tell me how to prove that in $SL_2({\mathbb{F}_q})$ the only element of even order is $-I$ ($ \ I$ - identity matrix)? I would really appreciate a thorough explanation, because I cannot ...