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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Discriminant of Polynomials (Galois Theory)

So I'm reading Dummit and Foote and they define the discriminant of $x_{1},...,x_{n}$ by $$D=\prod_{i<j}(x_{i}-x_{j})^2$$ and the discriminant of a polynomial to be the discriminant of the roots. ...
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Alternative way to count the number of solutions to the equation $x^2 + y^2 = -1$ over $\Bbb Z /p$

$x^2 + y^2 = -1$ is a weird equation because it has no solutions over $\Bbb R$. I want to count the number of solutions it has over $\Bbb Z / p$ where $p$ is prime. If $p = 2$ then it has $p$ ...
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How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any ...
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34 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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Counting the number of solutions of equation $x^2 + y^2 = 1$ over $\Bbb Z/p$

List proofs of the fact that the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$, where $p$ is a prime $\neq 2$, is $p-(-1)^{\frac{p-1}2}$. I thought of two. I write one below.
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55 views

From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
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43 views

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...
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20 views

find the prime factorization of $x^3-5x^2+6x+7$ in $Z/11Z$

I need to find the prime factorization of $f = x^3-5x^2+6x+7$ in $Z/11Z$ I tried the following but not sure if it is correct and if there is a better and faster way to do it. first i tried one by ...
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34 views

Sum of squares in finite field cannot be congruent to $0$? [closed]

Consider a finite field $GF(p)$, where p is a prime integer and $p\equiv 3 (mod 4)$. Consider two elements $a,b\in GF (p)$. How to prove $a^2+b^2\neq 0 (mod p)$.
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$\alpha \in \overline{\mathbb{F}}_q$ satisfying $\alpha^{q+1}+\alpha=-1$

Let $\overline{\mathbb{F}}_q$ be the algebraic closure of $\mathbb{F}_q$. Assume that $\alpha \in \overline{\mathbb{F}}_q$ satisfies at $$\alpha^{q+1}+\alpha=-1$$ Show that $\alpha \in ...
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Why does the plot of the legendre symbol of $x^2 - y^2$ over a finite field look rectangular

The small top-left thing is a plot of the legendre symbol of $x^2 - y^2$ over $\Bbb F_{37}$. The thing in the middle is plot for $\Bbb F_{587}$. The thing on the right is a plot of the legendre ...
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52 views

finding all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$, faster method?

I want to find all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$ and I wonder if there's a better method than the following. The polynomials are of form $aX^2 + bX +c$. So I have to ...
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2answers
40 views

Injectivity and norm function on finite fields [closed]

Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by $$f(x)=\alpha x^q+\alpha^q x$$ for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 ...
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35 views

Identification of two finite fields

I got the assignment to determine the identification of $\mathbb{F}_3^2$ and $\mathbb{F}_{3^2}$. I am capable of constructing the non-prime fields $\mathbb{F}_{3^2}$ as a reduction of $\mathbb{F}_3$ ...
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19 views

Roots of a polynomial over a finite field

Let $f(x)=a_0x+a_1x^q+... a_{k-1}x^{q^{k-1}}$ be a nonzero polynomial for a prime $q$. It is easy to observe that $$f:F_{q^n}\to F_{q^n}$$ a linear function. I want to show that $f$ has at most ...
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77 views

Finite fields and their subfields

Let $\mathtt{F}$ and $\mathtt{F'}$ be two finite fields of order $q$ and $q'$ respectively. Then: $\mathtt{F'}$ contains a subfield isomorphic to $\mathtt{F}$ if and only if $q\le q'$ ...
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Find the multiplicative inverse of $\mathbf{a+1}$ in $\mathbb{Z_2}(\mathbf{a})$

Q:- Let $\mathbf{a}$ be a zero of $x^3+x^2+1$ in some extension of $\mathbb{Z}_2$. Find the multiplicative inverse of $\mathbf{a+1}$ in this extension. Attempt: we know that if $F$ is a field and ...
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78 views

Newton's identities over finite fields

The Newton identities (including over finite fields) are given by $$ ke_k = \sum_{i=1}^k (-1)^{i-1} e_{k-i}p_i, $$ where the $e_k$ is the $k$-th elementary symmetric polynomials and the $p_k$ is the ...
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3answers
65 views

Is every finite field a quotient ring of ${Z}[x]$?

Is every finite field a quotient ring of ${Z}[x]$? For example, how a field with 27 elements can be written as a quotient ring of ${Z}[x]$?
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1answer
35 views

Square root of $-1$ over a finite field [duplicate]

It is known that the equation $x^2 \equiv -1 \pmod{p}$, where $p$ is an odd prime number, has a solution iff $p = 4k +1$ for some natural $k$. Does it exist a similar characterization for a general ...
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36 views

Describe all $p^{n}$ (in terms of congruence conditions of $p$ and $n$) for which $x^{2}+1$ irreducible over $\mathbb{F}_{p^{n}}$.

So I've said $x^{2}+1$ is reducible over $\mathbb{F}_{p^{n}} \iff \mathbb{F}_{p^{n}}$ contains a root $\alpha$. Hence if $\alpha$ is such a root then $\alpha^2 = -1$ so that $\alpha^4 =1$ and hence ...
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Shortest irreducible polynomials over $\Bbb F_p$ of degree $n$

For any prime $p$, one can realize any finite field $\Bbb F_{p^n}$ as the quotient of the ring $\Bbb F_p[X]$ by the maximal ideal generated by an irreducible polynomial $f$ of degree $n$. By dividing ...
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I need help with constructing the field GF$(2^8)$.

I have to use the polynomial $X^8 + X^4 + X^3 + X^2 +1$, and given $\alpha$ is a primitive element. So far I have that $\alpha^8=\alpha^4 + \alpha^3+\alpha^2+1$, and the only method for finding the ...
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32 views

How many quadratic non-residues are there in the image of $f(a) = a^2 - x^2$?

When reading about Cipolla's algorithm on Wikipedia, I found that the number of quadratic non-residues in the image of $f:\Bbb F \longrightarrow \Bbb F, f(a) = a^2 - x^2$ where $\Bbb F$ is a finite ...
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75 views

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ [duplicate]

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ Attempt. Since $x^2+x+1$ is of degree 2, it is reducible iff it has a root in $\mathbb{F}_p$. It has a root in ...
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Understanding an Opening to Finite Fields

While I have an understanding of some notions one learns in abstract algebra, that knowledge is extremely overshadowed by what I know in analysis. I opened up Jean-Pierre Serre's A Course in ...
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In a vector space over a finite field, can the orbit of a point under matrix multiplication have dependent subsets?

Let $\mathbb{F}_q^n$ be the vector space of dimension n over the finite field of order q, $\vec{v}$ a vector in the space, and $M$ an invertible $n \times n$ matrix over $\mathbb{F}_q$. We know that ...
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Proof remainder of polynomial division GF(2) can be calculated by LSFR

I have been reading that CRC, which is the calculation of the remainder of $x/P(x)$ in GF(2) can be implemented with a Linear Shift Feedback Register. However, I can't find the proof for this, or ...
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68 views

Constructing a multiplication table for a finite field

Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$ and let $F=\mathbb{Z}_2(\alpha)$, where $\alpha$ is a root of $f(x)$. Show that $F$ is a field and construct a multiplication table for $F$. Can you please ...
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If $f(x)=x^{m}+1$ is an irreducible polynomial in $\mathbb{F}_{p}[x]$, then prove that $2m$ divides $p^{m}-1$

Here's the full problem: Let $\mathbb{F}_{p}$ denote the finite field of size $p$, where $p$ an integer prime greater than $2$. Suppose that $f(x)=x^{m}+1$ is an irreducible polynomial in ...
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Field extensions for polynomial $T^3+2T +1$ in $\mathbb F_3$

I have the following polynomial over $\mathbb F_3$: $$ f(T) = T^3+2T+1 $$ I would like to find out a field extension in order to add the roots of this polynomial. Edit: by defining $\alpha\not\in ...
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50 views

Is the polynomial $x^8+x+1$ irreducible in $\mathbb{F}_2[x]$?

Is the polynomial $f(x)=x^8+x+1$ irreducible inf $\mathbb{F}_2[x]$? I know that if $x^8+x+1$ divides $x^{2^8}-x=x^{256}-x$, then it is irreducible over $\mathbb{F}_2$. I started using the division ...
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Period notation in group theory

In the context of finite fields, quotient groups, and characteristics, what does $$n.1=0$$ mean, i.e., what is the period notation?
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Extending Semilinear Transformations over Finite Fields

Suppose $ p $ a prime integer and $ m $ and $ n $ positive integers, where $ m | n $. Let $ \Phi_{m} $ denote the Frobenius automorphism of $ \mathbb{F}_{p^m} $ and $ \Phi_{n} $ the Frobenius ...
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Find a primitive element for the field extension $\mathbb{F}_{125}/\mathbb{F}_5$ - which elements of $\mathbb{F}_{125}$ are not in $\mathbb{F}_5$?

I want to find a primitive element for the field extension $\mathbb{F}_{125}/\mathbb{F}_5$. I constructed $\mathbb{F}_{125}$ as $\mathbb{F}_5[X]/\langle X^3 + X + 1 \rangle$. Since the degree of the ...
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26 views

Construction of non-prime finite fields

I am new to Galois field theory and I am struggling with some definitions. To construct any non-prime finite field $GF(p^n)$ with p prime and $n \in \mathbb{N}$, one has to find an irreducible ...
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Is it possible to define a continuous field with characteristic $\neq 0$?

For example, defining an addition and multiplication on the unit circle in the complex plane such that it forms a field. This would be a sort of continuous analog of the finite fields. Another way I ...
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Show that every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$

A (probably simple) question I encountered but I don't know how to approach: Let $K$ be a field of prime characteristic $p>0$. Show every $f(x) \in K[x]$ can be represented as $g(x^{p^e})$ ...
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Constructing binary codes and n-bit words of given length over given finite fields.

Given $(x_1,x_2,...,x_n)\in\Bbb F_{2}^n$, how can I construct a linear binary code $C$ of length $l$. Then construct a $y-bit$ code word $\in C$? Then later on generate a parity check matrix and a ...
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the degree of every irreducible polynomial that divides $x^p-x-a$ is the same.

let $F$ be a field with char$(F)=p>0$ where $p$ is a prime.given $a\in F^\times $ ($a\not=0$) denote \begin{equation*}f(x)=x^p-x-a\end{equation*} I'm trying to prove that the degree of every ...
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29 views

Number of points on an elliptic curve and it's twist over $\mathbb{F}_p$.

I have another probably very trivial question about elliptic curves. This wikipedia article gives the following formula $|E|+|E^d|=2p+2$ where $E$ is an elliptic curve over $\mathbb{F}_p$ and $E^d$ is ...
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1answer
44 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
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Show that $\mathbb{F}_q$ is a Splitting Field for Polynomial $x^q-x$

I have been given a homework problem which asks me to assume that $\mathbb{F}_q$ is a field of order $q$, and to show that $\mathbb{F}_q$ is a splitting field for the polynomial ...
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124 views

Construction of addition and multiplication table for GF(4)

I am dealing with finite fields and somehow got stuck. The construction of a prime field $GF(p), p \in \mathbb{P}$ is pretty easy because every operation is modulo p. In other words $GF(p)$ contains ...
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47 views

Regarding the structure of certain minimal polynomials

My question is: is the following statement true? If so, how might I go about proving it? Let $m$ be an integer not less than 1, let $F = GF(2^{6m})$, and let $L = GF(2^{2m})$. Let $\gamma$ be a ...
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30 views

Cyclic consecutive zeros of binary sequence with prime length

I found a feature that if $N>5$ is a prime, and $M \triangleq \frac{N-1}{2}$ is also a prime, then we will always have a binary sequence $x_1,\ldots, x_N$ with $L=\frac{N-1}{2}$(or ...
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73 views

Subgroups of automorphisms of Finite fields

Let $G$ denote the group of all the automorphisms of the Field $F_{3^{100}}$.Then,what is the number of distinct subgroups of $G$? First of all I have to compute $G$. Now \begin{equation*} ...
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1answer
11 views

Factorizing polynomials: How to calculate $g_{(X)}\in F_{q}[x]$ if we have $f_{(x)} = g_{(x)}^p$

How do I calculate $g_{(X)}\in F_{q}[x]$ if we have $f_{(x)} = g_{(x)}^p$ and $p$ is the characteristic of the field $F$? This problem arises from the factorization of a polynomial into irreducible ...
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57 views

Existence of Jordan decomposition over finite field

Prove that over finite field $\mathbb F$ exists additive Jordan-Chevalley decomposition: for all matrix $M$ there are semisimple matrix $M_{s}$ and nilpotent matrix $M_{n}$ such that $M=M_{s}+M_{n}$. ...
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1answer
69 views

How to find minimal polynomial of primitive element (field theory)

I am given a primitive element $\alpha$ in the Galoisfield $F_{2^6}$. The question is to find the mimimal polynomial of $\alpha^7$. How to I find this? My thoughts so far: $$ \alpha^7 \rightarrow ...