Tagged Questions

1
vote
1answer
25 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 ...
3
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2answers
39 views

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 ...
-1
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0answers
40 views

A question about hyperplanes in affine geometries [closed]

List all hyperplanes in $\operatorname{AG}_3(2)$ $\operatorname{AG}_4(2)$ What is the main idea while listing? Can you explain please?
3
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5answers
57 views

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 + ...
1
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1answer
38 views

Number of solutions of $P(x_1, …, x_n) = 0$ in $\mathbb{F}_q^n$

I have this exercise : $q = p^r$ (it is not clearly precised in the exercise that $r = n$). and $\mathbb{F}_q$ is the finite field of cardinal $q$. Let's $P \in \mathbb{F}_q[X_1, ..., X_n]$ of degree ...
2
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2answers
132 views

The linearity of the extended code

If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is: If the code $C$ is linear, can we prove that the extended code $C'$ is linear too? ...
1
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1answer
55 views

extended linear codes over the field $\mathbb F_q$

Suppose we extend the $[n,k]$ linear code $C$ over the field $\Bbb F_q$ to the code $C'$, where $$ C' = \{(x_1,\ldots ,x_n,x_{n+1})\in \Bbb F_q^{n+1} : (x_1,\ldots,x_n) \in C \text{ and } ...
1
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1answer
33 views

Quotient Space over finite field.

I'm looking at a vector space $V = F^3$ where $F = \{0,1,2\}$ the field with three elements. I have a subspace $W = \text{span}(1,2,1)$ and I'm trying to explicitly describe the quotient space $V/W$. ...
4
votes
1answer
131 views

Monic Irreducible Polynomials over Finite Field

Let $F=\mathbb{F}_{q}$ be a finite field (so $q=p^k$ for some prime $p$ and positive integer $k$), and let $\varphi(d)$ denote the number of monic irreducible polynomials of degree $d$ in $F[X]$. I'm ...
5
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1answer
142 views

Calculating Splitting Field Degree of Extension

Is there an easier way to calculate the degree of extension of a splitting field for a polynomial like $$x^7-3\quad\text{over}\quad\Bbb Z_5\,?$$My approach for several of these have been to find all ...
1
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1answer
145 views

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$

Determine the number of irreducible monic polynomials of degree 3 in $\mathbb F_p[x]$ where $p$ is a prime. I'd like to start off by acknowledging that I know there are many posts relating to similar ...
0
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1answer
67 views

Roots of unity in a field generated by a root of a polynomial

The polynomial x^3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root k to make the field F := F7(k), which will be of degree 3 over F7 and therefore of size 343. The ...
2
votes
1answer
109 views

Using characters in finite fields to find number of solutions to polynomials.

I am trying to use the theorem below to show that if $d_i=(m_i,p-1)$ then $\sum_ia_ix_i^{m_i}=b$ and $\sum_ia_ix_i^{d_i}=b$ have the same number of solutions. So far, I have been able to prove that if ...
0
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1answer
83 views

Nonzero trace in finite fields and proving irreducibility.

If we define trace to be $x+x^p+\cdots+x^{p^{n-1}}$. How do we know there is an element of nonzero trace? Clearly if $a\in F_p$ then its trace is zero as $a^{p^i}=a$ so ...
0
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1answer
78 views

Finite Fields of Order 3

I have been trying to learn Real Analysis, though I'm having trouble with a problem. Show that there exists one and (essentially) only one field with three elements. Any help will be appreciated.
1
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1answer
97 views

Number of elements in a finite field extension for finite fields

Given an arbitrary finite field $K$ (not necessarily $\mathbb{F}_p$ with $p \in \mathbb{P}$) with $|K| = q$ and an irreducible polynomial $f$ with $\alpha$ as root and degree of $n$. Is $|K(\alpha)| = ...
0
votes
4answers
113 views

Roots in a finite field

Given a finite field $|K|=q$ and an irreducible $f \in K[x]$ with $\deg(f)=n$ with $\alpha$ as a root. My candidates for the roots are $\alpha, \dots , \alpha^{q^{n-1}}$. Assuming $\alpha^{q^i} = ...
7
votes
3answers
596 views

Galois Group over Finite Field

I am having a bit of difficulty trying to answer the following question: What is the Galois group of $X^8-1$ over $\mathbb{F}_{11}$? So far I have factored $X^8-1$ as ...
2
votes
1answer
153 views

Field Extension of $F_p(X^p,Y^p)$

Let $K=F_p(X,Y)$, where $F_p$ is a finite field of characteristic $p$, and $F=F_p(X^p,Y^p)$. I have been given the following problem: Determine the degree of extension $[K:F]$. My experience with ...
2
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1answer
70 views

Equation of the line in an affine plane over a polynomial field

What are some examples of this? Say for $F_{4}$. I know this is a very simple question, but I can't find any info on it. Edit: Yes, I was thinking of $F_{2}[x]/(x^2+x+1)$. I was confused.
0
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2answers
284 views

Quotient Rings of Polynomials Over Finite Fields

I have this question which I don't know how to approach: Let ${F}_{2} = {Z}/2Z$, find representatives for the residue classes of ${F}_{2}[X]$ modulo the polynomial $f(x)$ and compute the ...
1
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1answer
180 views

Permutation matrix to “invert” part of a matrix?

I am working with a Linear Code, C, over $F_{2^5}$, and its dual, $C^{\perp}$. I have the generator matrix for $C$, $G$, and have calculated the generator matrix for $C^{\perp}$, $G^{\perp}$. I need ...
1
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1answer
153 views

Product in GF(16)

i need some help with a product in GF(16), where it is seen as an extension of GF(4)={0, 1, x, x+1} (where $x^2 = x + 1$ ) with the irreducible polynom $f(y) = y^2 + y + x$ So the elements in the ...
5
votes
1answer
183 views

Is correcting 2 consecutive error's in 9 messages from $ GF(2^6) $ by turning tham into 3 messages and solving Reed-Solomon code $ (3, 18) $ possible?

2 consecutive messages have errors. We have 9 messages from $ GF(2^6, x^6+x+1) $. Messages were encoded with $ (x+1)(x+a)(x+a^2)\sum_{i=1}^6X_ix^{i-1}=\sum_{l=1}^9Y_lx^{l-1} $ , where $ ...
1
vote
1answer
111 views

How to decode encoded, and corrupted in transmission message in Galois Field $2^5$ with one error?

We are given $GF(2^5, x^5+x^2+1 )$. We had some $ X_1, ..., X_5 $ message items from our $ GF(32) $ which we do not know and need to find. Thay were encoded via service blocks $ Y_1...Y_7 $ with next ...
0
votes
3answers
110 views

finite fields factorization

Let $\mathbb{F}_2$ be the finite field with two elements. Let $f(x) = x^6+x^4+x+1$ be in $\mathbb{F}_2[x]$. If $f(x)$ is irreducible, give a reason. If it is not irreducible, determine a factorization ...
2
votes
3answers
96 views

Count points and lines in $\mathbb{A}^2(\mathbb{F}_p)$

Let $p$ be a prime, then $\mathbb{F}_p$ is a finite field. $\mathbb{A}^2(\mathbb{F}_p)$ is an affine plane. Number of points in $\mathbb{A}^2(\mathbb{F}_p)$ is $p^2$. I look at a line equation ...
1
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1answer
242 views

Numbers of vectors in a vector space over a finite field, with different multiplication

I had a recent question in an assignment that I couldn't complete. We are given the following: $q$ is an odd prime power. $(F,+,\cdot)=\text{GF}\left(q^2\right)$. $K$ is the $q$ element subfield of ...
3
votes
2answers
225 views

For which prime $p$ are the additive groups $\mathbb{F}_p$ and $\mathbb{F}_p^2$ $\mathbb{Z}[i]$-modules?

Homework for my algebra class. Chapter 14, Exercise 7.8 in Artin's Algebra, Second Edition: Let $F = \mathbb{F}_p$. For which prime integers $p$ does the additive group $F^1$ have a structure ...
2
votes
2answers
164 views

Using Carlitz's exponential formula to prove an identity

This is a question on the homework for my finite fields class. The beginning of the assignment defines the following notation: For $i\geq1 $, define the following elements of $A=\mathbb{F}_{q}[T]$ : ...