# Tagged Questions

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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### matrix rank over finite field

Let $M$ be a square matrix over finite field $\Bbb F_p$. Let $N$ be the matrix over $\Bbb F_p$ obtained replacing every non-zero entry of $M$ by $1$. Is $Rank(N)\leq Rank(M)^{f_p}$? for some constant ...
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### GCD of polynomials in $\mathbb{F}_2[x]$

How can one show that $\gcd(x^4+x+1,x^4-x)=1$ in $\mathbb{F}_2[x]$? Is it because the polynomials do not share any irreducible monic polynomials i.e. $1, x, x+1$ as factors?
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### Splitting field of $x^ {a^n}$ −1 in Z/aZ[x]

What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural? I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a ...
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### Help with problem of the book Algebra of T. Hungerford

This is a problem in “Algebra” by T. Hungerford: If $|K|=q$ and $f\in K[x]$ is irreducible, then $f$ divides $x^{q^n}-x$ if and only if $\deg f$ divides $n$. I found it difficult to solve ...
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### Explanation of division/reduction in a binary Galois Field using bit-shifts

I've seen a lot of algorithms reducing the result of a multiplication in a Binary Field by using only bit-shifts and XOR. The number of positions to shift seems to be derived from the polynomial, but ...
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### Product of random binary vector with random binary matrix in GF(2)

Suppose we have a binary vector $f$ with dimensions $1×l$ such that each entry in the vector is generated independently with propability $q$ of being $1$. And we have a binary matrix $G$ with ...
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### Show that a matrix is not diagonalizable over a finite field

I have a matrix: $$\ \left[ {\begin{array}{cc} 2 & 2 \\ 1 & 2 \\ \end{array} } \right]$$ which I need to show that it cannot be diagonalized over the finite field $\mathbb{F_3}$. ...
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### How to find all of generators in a finite fields

How can I find all generators of a finite field? For example in GF(2^3) and X^3 + x^2 + 1 as primitive polynomial. I don`t want all of solutions. I need some hint and help to solve this problem. ...
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### If $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field , then $T^n$ is the identity operator?

If $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field , then is it true that $T^n = I$ ( the identity operator) for some positive integer $n$ ?
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### Number of roots of a polynomial over a finite field

For any $g$ in $\mathbb{Z}/p\mathbb{Z}[x]$ prove that the degree of $f = \gcd(x^p - x, g(x))$ is exactly the number of distinct roots of $g$ in $\mathbb{Z}/p\mathbb{Z}$. My main problem is that I do ...
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### Splitting field in finite field

What is the splitting field of the polynomial $X^{p^8}-1$ over $\mathbf F_p$? I'm confused, is not $X^{p^8}-1=(X-1)^{p^8}$ then the splitting field is $\mathbf F_p$? Thanks.
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### Polynomial Factoring over a finite field

Ok, so I'm trying to figure out how to factor polynomials over a finite field. My polynomial is x^5 + x^2 + x + 1 and I have to factor over GF(2) I know the answer is (x+1)^2 * (x^3 + x + 1), because ...
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### Find the number of primitive elements

How can i find the number of primitive elements over the field of order q? GF(27) for example. Is there a formula that I can follow? I'm really confused on how to find them. Any help would be much ...
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### an irreducible polynomial over GF(2) is primitive over GF(2)

let $P \in F_{2} [X]$ of degree $7$, how to prove this: P is irreducible $\Leftrightarrow$ P is primitive i tried to use the mersenne prime !
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### Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
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### When Errors Go Undetected in CRC

I understand that CRC will not be able to detect errors if: The remainder of $E(x) / G(x) = 0$ $E(x) = G(x).Z(x)$ for some polynomial $Z(x)$ I understand the first point, which means that if the ...
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### Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
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### Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating $N_s$...
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### polynomial over finite field, roots forming additive subgroup

Let $q=2^h$ and $t=2^r$ for some $h\ge r$ and denote by $\mathbb{F}_q$ the finite field of order $q$. (since the previous, simple version was wrong, I'm posting here a new version) Let $f$ be a ...
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### Question about the Characteristic of $\mathbb{F}_{p^n}$

We can prove that any finite field of prime characteristic $p$ must have $p^n$ elements. Conversely, let $F$ be a finite field with $p^n$ elements, where $p$ is a prime number. Is the following ...
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### Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
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### How to prove a finite field is not ordered?

I have a set S={0,1}, and the addition and multiplication rules are \begin{array}{c|cc} +&0&1\\ \hline 0&0&1\\ 1&1&0 \end{array} \begin{array}{c|cc} *&0&1\\ \hline 0&...
Let $\mathbb{F}_4=\{0,1,u,u^2\}$ be the field with $4$ elements. Is there a polynomial $p \in \mathbb{F}_2[x,y]$ with the following property? (1) For $r,s \in \mathbb{F}_4$, we have $p(r,s)=u \... 4answers 36 views ### Calculating$3/10$in$\mathbb{Z}_{13}$I'm trying to calculate$\frac{3}{10}$,working in$\mathbb{Z}_{13}$. Is this the correct approach? Let$x=\frac{3}{10} \iff 10x \equiv 3 \bmod 13 \iff 10x-3=13k \iff 10x=13k+3$for some$k \in \...
Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4-x$ to $\mathbb Z/2\mathbb Z$. Definition: A polynomial $f(x)\in F[x]$ splits over $F$ if it is ...