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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Count the number of monic irreducible polynomials of degree 12 over $\mathbb F_q$

This is a qualifying problem. I cannot understand how the inclusion exclusion principle work here in detail. However, I have an argument which leads to a different answer. I am not sure ...
4
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33 views

Number of Solutions to Polynomials in Finite Fields

Let $\mathbb{F}$ be a finite field and $f_i\in\mathbb{F}[x_1,x_2,\ldots,x_n]$ be polynomials of degree $d_i$, where $1\leq i\leq r$, such that $f_i(0,\ldots,0) = 0$ for all $i$. Show that if ...
2
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1answer
44 views

Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$, for a fixed $p$. I will consider only prime fields, not $GF(p^n)$. Represent the $p$ elements of the field as integers $\{0,1,\ldots ...
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44 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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38 views
+100

How to attack universal hash function based on finite-field arithmetic?

As per the Recursive n-gram hashing is pairwise independent, at best paper, I want to use the algorithm described in chapter 6 and 7 (page 7 - 10). The hash works as follows: Define a random ...
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2answers
33 views

How to find orthogonal vectors in GF(2)

I've 13 rows in a matrix, which are linearly independent.(number of columns is 20), in GF(2). Now i have to find 20 orthogonal vectors in GF(2). I've added 20 more rows which are the rows of an ...
2
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1answer
19 views

Unit group of a field is divisible

In the lecture notes on Valuation theory, in Ex. $1.16$ on page $11$ we are asked to show that: If $k$ is an algebraically closed field, then $k^{\times}$ is a divisible abelian group. Isnt $k = ...
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105 views
+50

Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
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135 views

How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
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0answers
18 views

Computing number of irreducible polynomials of degree n over $\mathbb{F}_q$

When I try to find the number of irreducible polynomials (of degree n) over a finite field I first look for the number of $\alpha \in \mathbb{F}_{q^n}$ such that ...
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1answer
19 views

I have to show it is isomorphic to $K = GF(p^{kd})$ [closed]

Suppose $F = GF(p^k)$ is a finite field. I know $F[C]$ is a field extension of $F$ with degree $d = \deg m$, and I have to show it is isomorphic to $K = GF(p^{kd})$ (where $C$ is a companion matrix ...
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138 views

Four questions about finite fields

Is $\mathbb{F}_5$ a subfield of $\mathbb{F}_7$? I can think of the answer 'yes' because they have the same set op operations $+ \cdot$ and the answer 'no' because in $\mathbb{F}_5: 2\cdot3=1$ and in ...
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3answers
25 views

Finding an isomorphism between polyomial quotient rings

Let $F_1 = \mathbb{Z}_5[x]/(x^2+x+1)$ and $F_2 = \mathbb{Z}_5[x]/(x^2+3)$. Note neither $x^2+x+1$ nor $x^2+3$ has a root in $\mathbb{Z}_5$, so that each of the above are fields of order 25, and hence ...
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1answer
41 views

How does the multiplicative group of a finite field, considered as a vector space, act on subspaces?

Given that a finite vector space $V = \operatorname{GF}(p)^n$ corresponds to the finite field $F = \operatorname{GF}(p^n)$, I'm wondering about how the multiplicative subgroup of $F$, $F^*$, acts on ...
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1answer
23 views

What is the exponent in the definition of a Galois field called?

From what I understand, when speaking of a Galois field $\operatorname{GF}(p^k)$, $p$ is called the characteristic of the field, and $p^k$ is the order. Does $k$ have a name by itself?
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2answers
113 views

Modular curves over finite fields

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
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1answer
34 views

List all elements in the residue field $Z[i]/(q)$

Consider a Gaußian prime $q$. How to list all elements in the residue field $Z[i]/(q)$? Is there any formulas or criteria? Here I'm looking for the case $q$ is a complex number, as I can do the real ...
2
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2answers
55 views

Proof that algebraically closed fields of characteristic $p$ exist

How do you prove that algebraically closed fields of characteristic $p$ exist? I have also read: For a finite field of prime power order $q$, the algebraic closure is a countably infinite field ...
2
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1answer
26 views

Sum of powers in finite fields

I have trouble following the logic in this proof. In particular, why is the following equality is true: $$\displaystyle\sum_{x \in K^\times} x^u = \displaystyle\sum_{x \in K^\times} y^u x^u$$
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88 views

Polynomials over a finite field

Let $\mathbb{F}_p$ be a finite field where $p$ is a prime. Consider the following set of polynomials over $\mathbb{F}_p$: $$G_n(p)=\{{x+a_2x^2+\cdots+a_nx^n\mid a_i\in \mathbb{F}_p}\}.$$ Is ...
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1answer
45 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
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1answer
48 views

Is finding generators of finite fields hard?

Task: Given $n$, find a generator of $GF(n)^*$. Is there any evidence this is hard? Maybe a reduction from another problem presumed hard? Finding the orders of elements should be hard because I ...
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102 views

Existence of ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field

Does a ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field with characteristic $p\equiv 3 \bmod 4$ such that the unity is mapped onto the unity exist? Thank for your help.
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47 views

Some questions on elliptic curves over finite fields

Let $E$ be an elliptic curve defined over $\mathbb{F}_q$. For a prime $\ell \neq q$, we have that the $\ell$-torsion subgroup $E[\ell] \cong (\mathbb{Z}/\ell \mathbb{Z})^2$. As can be easily seen, ...
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30 views

Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes of ...
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2answers
55 views

Why are these curves' points called rational?

While studying curves defined over finite field $\mathbb F_q$, it's said that the points of the curve are the rational points. Why is it said like this? For example, if $q$ is prime, aren't we just ...
2
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1answer
39 views

A count involving number of subspaces

Let $V$ be an $n$ dimensional vector space over a finite field with $q$ elements and $W$ be a fixed $k$ dimensional subspaces of $V$. How to find the number of distinct subspaces $X$ of $V$ such that ...
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3answers
42 views

List the elements of the field $K = \mathbb{Z}_2[x]/f(x)$ where $f(x)=x^5+x^4+1$ and is irreducible

Since $\dim_{\mathbb{Z}_2} K = \deg f(x)=5$, $K$ has $2^5=32$ elements. So constructing the field $K$, I get: \begin{array}{|c|c|c|} \hline \text{polynomial} & \text{power of $x$} & ...
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22 views

FSR function of the component-wise product, sum, of two LFSR sequences

Let $T_1$, $T_2$ be two $m$-sequences over $\mathbb{F}_q$ of length $q^n-1$, say $T_1 = (\text{Tr}_{q^n | q}(\alpha^i))_{i \geq 0}$, $T_2 = (\text{Tr}_{q^n | q}(\beta^i))_{i \geq 0}$, for some ...
3
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1answer
57 views

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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38 views

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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0answers
32 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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2answers
95 views

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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1answer
50 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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1answer
40 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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1answer
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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0answers
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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1answer
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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 ...
2
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2answers
39 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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0answers
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 ...
4
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1answer
73 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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2answers
72 views

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 ...
3
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66 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
57 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 ${Z}[x]$?
1
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1answer
34 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 ...
0
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1answer
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 ...