3
votes
2answers
54 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 $u \in F^*$ ...
2
votes
2answers
43 views

$F$ 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 ...
3
votes
1answer
48 views

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 ...
3
votes
2answers
74 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 ...
4
votes
4answers
81 views

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 ...
0
votes
1answer
19 views

Prove that the image $\alpha$ of $X$ in $\left(\mathbb{F}_3[X]/(X^3-X^2+1)\right)^{\times}$ is a generator

I'm trying to do an exercise of my homework that sais I have to prove that the iamge of $X$ in $K^{\times}=\left(\mathbb{F}_3[X]/(X^3-X^2+1)\right)^{\times}$ is a generator. Acording to what I know, ...
0
votes
1answer
22 views

Minimum polynom of an element in $K=\mathbb{F}_5[x]/(x^2-2)$

I want to know how to calculate the minimum polynom of an element $\alpha$ in $K=\mathbb{F}_5[X]/(X^2-2)$ where $\alpha$ is the image on $K$ of $X+2$ I'm already verficated that $K$ is a field. As I ...
2
votes
2answers
37 views

Finding the kernel and image of a linear transformation over the field $\Bbb Z_2$

Given the A vector space $V$ over the field $\Bbb Z_2$ and a linear map $t:V \to V $ .following matrix $T =\begin{bmatrix}3& -1 &1\\-1 & 5 & -1\\ 1 & -1 & 3\end{bmatrix}$ ...
1
vote
1answer
41 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
3
votes
1answer
49 views

Factoring $x^{16}-x$ over $\mathbb{F}_8$

A homework question asks me to factor $x^{16}-x$ over the finite fields $\mathbb{F}_4$ and $\mathbb{F}_8$. I got the result for $\mathbb{F}_4$ using the factoring over $\mathbb{F}_2$ and then a ...
0
votes
0answers
23 views

Prove that if $f$ is a primitive polynomial over $F_q$ then $f$ divides $Q_{q^m-1}$.

I am not writing my complete proof, and my conclusion is that since all the roots of $f$ are primitive $(q^m-1)st$ roots of unity and so are the roots of $Q_{q^m-1}$. Therefore, $f$ must divide ...
9
votes
1answer
133 views

Proving $f = x^8+x^7+x^3+x+1$ to be irreducible over $\mathbf{F}_2$.

I am taking all the irreducible polynomials over $\mathbf{F}_2$ of degree 1 to 4 inclusive and using the long division to determine whether the given $f$ is divisible by anyone of them and it is not ...
1
vote
1answer
143 views

Irreducible polynomial f in polynomial ring of finite field order n divides a particular polynomial if and only if degree of f divides n

Here is the problem I am attempting: Suppose that $K$ is a finite field with $|K| = q$. Show that if $f(x) \in K[x]$ is irreducible, then $f$ divides $g(x) = x^{q^n}-x$ in $K[x]$ if and only if ...
1
vote
1answer
36 views

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors?

What is the smallest Finite Field in which the following polynomial is factorizable to irreducible factors? $$(x^2+x+1)(x^5+x^4+1)(x^7+x^6+x^3+1) $$
1
vote
1answer
58 views

Addition in finite fields

For a question, I must write an explicit multiplication and addition chart for a finite field of order 8. I understand that I construct the field by taking an irreducible polynomial in $F_2[x]$ of ...
1
vote
0answers
69 views

On the fields of rational fractions over $\mathbb{F}_{p}$

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of ...
1
vote
1answer
57 views

Counting elements of $y^2 - y = x^3$ in finite fields

The problem I have to solve is the following: Let $p$ be a prime number with $p \equiv 2$ mod $3$. Let $E$ be the elliptic curve given by $y^2 - y = x^3$. Show that $\#E(\mathbb{F}_p) = p+1$ and ...
2
votes
3answers
113 views

Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements

I am attempting to prove what the title says, that $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements. I have already proven that for any field ...
0
votes
0answers
25 views

a is zero over Z${p}[x]$, $a^p$ is zero too if p is prime? [duplicate]

p is prime , Z${p}$ is a field with p elements and a $\epsilon$ K , K is a extension of Z${p}$ a) if q(x) is a polynom in Z${p}$[x] and a is a zero of q(x), Proof that is true that $a^p$ is zero of ...
2
votes
1answer
35 views

Prove that for two vectors x,y over GF(q), the number of vectors that are closer to x is the same as the number of vectors that closer to y.

Let $x,y\in\mathbb F_q^n$ be vectors. We'll define: $X= \{ u\in\mathbb F_q^n \mid d(x,u)<d(y,u)\}$ $Y= \{ u\in\mathbb F_q^n \mid d(y,u)<d(x,u)\}$ Prove that $|X|=|Y|$. Well. ...
6
votes
3answers
48 views

Find the minimal $n$ such that there exists $[n,n-5]$ cyclic binary code with generator polynomial $g(x)=1+x^4+x^5$

Find the minimal $n$ for there exists $[n,n-5]$ cyclic binary code with generator polynomial $g(x)=1+x^4+x^5$. I couldn't figure out the answer. The only way I could think of is find out all the ...
0
votes
2answers
463 views

Definition of a Field of Characteristic $n$?

Let $V$ be a vector space over a field of characteristic not equal to $2$. Prove that $\{u, v\}$ is linearly independent with $u, v$ being distinct if and only if $\{u+v, v-v\}$ is linearly ...
0
votes
2answers
96 views

Calculate a sum of elements over a finite field $\mathbb{Z}_n$

I have this question for homework that I cannot solve. For $n\ge5$ prime number, calculate $1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n-1)^2}$ in $\mathbb{Z}_n$. I tried proving it using ...
3
votes
2answers
90 views

Is $2+5x$ a primitive root in $\mathbb{F}_7[x]/(x^2+1)$?

The question I'm inquiring about is all in the title, but I would be more interested in a few things related to the question which I don't know. I know what a primitive root of $\mathbb{F}_p$ is for ...
4
votes
2answers
91 views

If $f\in\mathbb{F}_p[x]$ is irreducible and has a root in $\mathbb{F}_{p^n}$, then $f$ splits over $\mathbb{F}_{p^n}$ [duplicate]

Let $f(X) \in \mathbb F_p[X]$ irreducible with $p$ prime and assume $\exists \alpha \in \mathbb F_{p^n}: f(\alpha) = 0$ where $n \geq 1$. I then have to prove that $f$ splits over $\mathbb ...
1
vote
1answer
120 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 ...
5
votes
2answers
485 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 ...
3
votes
5answers
74 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
vote
1answer
43 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
votes
2answers
151 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
vote
1answer
82 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
vote
1answer
67 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 $V/W$. ...
5
votes
1answer
604 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
votes
1answer
275 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 ...
6
votes
1answer
322 views

Splitting field of $ x^2 + 1$ over $\mathbb{Z_3}$

I have the following exercise: Find splitting field for the polynomial $x^2 + 1$ over $\mathbb{Z_3}$. My solution: At first, we should try to solve the equation $x^2 + 1 = 0$, thus $x^2 = 2$ ...
2
votes
1answer
740 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
votes
1answer
156 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
118 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
votes
1answer
161 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
votes
1answer
161 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
vote
1answer
136 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
126 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} = ...
9
votes
3answers
1k 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
209 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
votes
1answer
85 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
votes
3answers
526 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
vote
1answer
231 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
vote
1answer
277 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 ...
6
votes
1answer
205 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
125 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 ...