# Tagged Questions

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### 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 $u \in F^*$ ...
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### $F$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 $T =\begin{bmatrix}3& -1 &1\\-1 & 5 & -1\\ 1 & -1 & 3\end{bmatrix}$ ...
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### 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 $ker(f)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? $$(x^2+x+1)(x^5+x^4+1)(x^7+x^6+x^3+1)$$
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 $x^2 = 2$ ...
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### 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 ...
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### Roots of unity in a field generated by a root of a polynomial

The polynomial x^3 âˆ’ 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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 ...